ContentsArticlesInteger sequence Sequence Integer Abundant number BaumSweet sequence Bell number Binomial coefficient Carmichael number Catalan number Composite number Deficient number Euler numbers Even and odd numbers Factorial Fibonacci number Fibonacci word Figurate numbers Golomb sequence Happy number Highly totient number Highly composite number Home prime Hyperperfect number Juggler sequence Kolakoski sequence Lucky number Lucas number Padovan sequence Partition number Perfect number Pseudoperfect number Prime number Pseudoprime Regular paperfolding sequence 1 3 7 11 12 13 19 35 39 48 49 50 51 54 68 85 88 90 91 96 97 100 101 106 107 109 111 113 117 125 129 130 146 147

RudinShapiro sequence Semiperfect number Semiprime Superperfect number Thue-Morse sequence Ulam numbers Weird number Recursion theory Definable set Countable Uncountable Cardinality Beth one Complete sequence

149 150 151 153 154 158 160 161 171 173 179 181 184 187

ReferencesArticle Sources and Contributors Image Sources, Licenses and Contributors 189 193

Article LicensesLicense 194

Integer sequence

1

Integer sequenceIn mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, is formed according to the formula n21 for the nth term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the nth perfect number.

ExamplesInteger sequences which have received their own name include: Abundant numbers BaumSweet sequence Bell numbers Binomial coefficients Carmichael numbers Catalan numbers Composite numbers Deficient numbers Euler numbers Even and odd numbers Factorial numbers Fibonacci numbers Fibonacci word Figurate numbers Golomb sequence Happy numbers Highly totient numbers Highly composite numbers Home primes Hyperperfect numbers Juggler sequence Kolakoski sequence Lucky numbers Lucas numbers Padovan numbers Partition numbers Perfect numbers Pseudoperfect numbers Prime numbers

Pseudoprime numbers Regular paperfolding sequence RudinShapiro sequence

Integer sequence Semiperfect numbers Semiprime numbers Superperfect numbers Thue-Morse sequence Ulam numbers Weird numbers

2

Computable and definable sequencesAn integer sequence is a computable sequence, if there exists an algorithm which given n, calculates an, for all n > 0. An integer sequence is a definable sequence, if there exists some statement P(x) which is true for that integer sequence x and false for all other integer sequences. The set of computable integer sequences and definable integer sequences are both countable, with the computable sequences a proper subset of the definable sequences (in other words, some sequences are definable but not computable). The set of all integer sequences is uncountable (with cardinality equal to that of the continuum); thus, almost all integer sequences are uncomputable and cannot be defined.

Complete sequencesAn integer sequence is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.

External links Journal of Integer Sequences [1]. Articles are freely available online. Inductive Inference of Integer Sequences [2]

References[1] http:/ / www. math. uwaterloo. ca/ JIS/ index. html [2] http:/ / www. cs. cmu. edu/ afs/ cs/ user/ mjs/ ftp/ thesis-program/ 2010/ theses/ tetruashvili. pdf

Sequence

3

SequenceIn mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function. For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence() is included in most notions of sequence, but may be excluded depending on the context.

Examples and notationThere are various and quite different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the notations introduced below. In addition to identifying the elements of a sequence by their position, such as "the 3rd element", elements may be given names for convenient referencing. For example a sequence might be written as (a1, a2, a2, ), or (b0, b1, b2, ), or (c0, c2, c4, ), depending on what is useful in the application.

An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy. It is bounded, however.

Finite and infiniteA more formal definition of a finite sequence with terms in a set S is a function from {1, 2, ..., n} to S for some n 0. An infinite sequence in S is a function from {1, 2, ... } to S. For example, the sequence of prime numbers (2,3,5,7,11, ) is the function 12, 23, 35, 47, 511, . A sequence of a finite length n is also called an n-tuple. Finite sequences include the empty sequence ( ) that has no elements. A function from all integers into a set is sometimes called a bi-infinite sequence or two-way infinite sequence. An example is the bi-infinite sequence of all even integers ( , -4, -2, 0, 2, 4, 6, 8 ).

Sequence

4

MultiplicativeLet A=(asequence defined by a function f:{1, 2, 3, ...} {1, 2, 3, ...}, such that a i = f(i). The sequence is multiplicative if f(xy) = f(x)f(y) for all x,y such that x and y are coprime.[1]

Types and properties of sequencesA subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. If the terms of the sequence are a subset of an ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This is a special case of the more general notion of monotonic function. The terms nondecreasing and nonincreasing are used in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively. If the terms of a sequence are integers, then the sequence is an integer sequence. If the terms of a sequence are polynomials, then the sequence is a polynomial sequence. If S is endowed with a topology, then it becomes possible to consider convergence of an infinite sequence in S. Such considerations involve the concept of the limit of a sequence. If A is a set, the free monoid over A (denoted A*) is a monoid containing all the finite sequences (or strings) of zero or more elements drawn from A, with the binary operation of concatenation. The free semigroup A+ is the subsemigroup of A* containing all elements except the empty sequence.

Sequences in analysisIn analysis, when talking about sequences, one will generally consider sequences of the form or which is to say, infinite sequences of elements indexed by natural numbers. It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by xn = 1/log(n) would be defined only for n 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given N. The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. Even more generally, one can study sequences with elements in some topological space.

Sequence

5

SeriesThe sum of terms of a sequence is a series. More precisely, if (x1, x2, x3, ...) is a sequence, one may consider the sequence of partial sums (S1, S2, S3, ...), with

Formally, this pair of sequences comprises the series with the terms x1, x2, x3, ..., which is denoted as

If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see series.

Infinite sequences in theoretical computer scienceInfinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as sequences or streams, as opposed to finite strings. Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0,1}). The set C = {0, 1} of all infinite, binary sequences is sometimes called the Cantor space. An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to 1 if and only if the n th string (in shortlex order) is in the language. Therefore, the study of complexity classes, which are sets of languages, may be regarded as studying sets of infinite sequences. An infinite sequence drawn from the alphabet {0, 1, ..., b1} may also represent a real number expressed in the base-b positional number system. This equivalence is often used to bring the techniques of real analysis to bear on complexity classes.

Sequences as vectorsSequences over a field may also be viewed as vectors in a vector space. Specifically, the set of F-valued sequences (where F is a field) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers. In particular, the term sequence space usually refers to a linear subspace of the set of all possible infinite sequences with elements in .

Doubly-infinite sequencesNormally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the otherthe sequence has a first element, but no final element (a singly-infinite sequence). A doubly-infinite sequence is infinite in both directionsit has neither a first nor a final element. Singly-infinite sequences are functions from the natural numbers (N) to some set, whereas doubly-infinite sequences are functions from the integers (Z) to some set. One can interpret singly infinite sequences as elements of the semigroup ring of the natural numbers doubly infinite sequences as elements of the group ring of the integers Cauchy product of sequences. , and

. This perspective is used in the

Sequence

6

Ordinal-indexed sequenceAn ordinal-indexed sequence is a generalization of a sequence. If is a limit ordinal and X is a set, an -indexed sequence of elements of X is a function from to X. In this terminology an -indexed sequence is an ordinary sequence.

Sequences and automataAutomata or finite state machines can typically be thought of as directed graphs, with edges labeled using some specific alphabet . Most familiar types of automata transition from state to state by reading input letters from , following edges with matching labels; the ordered input for such an automaton forms a sequence called a word (or input word). The sequence of states encountered by the automaton when processing a word is called a run. A nondeterministic automaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some input letter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence of single states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally used to mean the latter.

Types of sequences 1-sequence Arithmetic progression Cauchy sequence Farey sequence Fibonacci sequence Geometric progression Look-and-say sequence ThueMorse sequence

Related concepts List (computing) Ordinal-indexed sequence Recursion (computer science) Tuple Set theory

Operations on sequences Cauchy product Limit of a sequence

References[1] Lando, Sergei K.. "7.4 Multiplicative sequences". Lectures on generating functions. AMS. ISBN0821834819.

External links The On-Line Encyclopedia of Integer Sequences (http://www.research.att.com/~njas/sequences/index.html) Journal of Integer Sequences (http://www.cs.uwaterloo.ca/journals/JIS/index.html) (free) Sequence (http://planetmath.org/?op=getobj&from=objects&id=397) on PlanetMath

Integer

7

IntegerThe integers (from the Latin integer, literally "untouched", hence "whole": the word entire comes from the same origin, but via French[1] ) are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (1, 2, 3, ...). Viewed as a subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... 2, 1, 0, 1, 2, ...}. For example, 65, 7, and 759 are integers; 1.6 and 1 are not integers.Symbol often used to denote the set of integers

The set of all integers is often denoted by a boldface Z (or blackboard bold for Zahlen (German for numbers, pronounced integers modulo n (for example, ).German pronunciation:[tsaln]).

, Unicode U+2124 ), which stands[2]

The set

is the finite set of

The integers (with addition as operation) form the smallest group containing the additive monoid of the natural numbers. Like the natural numbers, the integers form a countably infinite set. In algebraic number theory, these commonly understood integers, embedded in the field Integers can be thought of as discrete, equally spaced points on an infinitely long of rational numbers, are referred to as number line. rational integers to distinguish them from the more broadly defined algebraic integers (but with "rational" meaning "quotient of integers", this attempt at precision suffers from circularity).

Algebraic propertiesLike the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative). The following lists some of the basic properties of addition and multiplication for any integers a, b and c.Addition Closure: Associativity: Commutativity: a+b is an integer Multiplication ab is an integer

a+(b+c)=(a+b)+c a(bc)=(ab)c a+b=b+a ab=ba a1=a An inverse element usually does not exist at all.

Existence of an identity element: a+0=a Existence of inverse elements: Distributivity: No zero divisors: a+(a)=0

a (b + c) = (a b) + (a c) and (a + b) c = (a c) + (b c) If ab = 0, then a = 0 or b = 0 (or both)

In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + ... + 1 or (1) + (1) + ... + (1). In fact, Z under addition is the only infinite cyclic group, in the sense

Integer that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However not every integer has a multiplicative inverse; e.g. there is no integer x such that 2x = 1, because the left hand side is even, while the right hand side is odd. This means that Z under multiplication is not a group. All the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. Adding the last property says that Z is an integral domain. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field. The smallest field containing the integers is the field of rational numbers. This process can be mimicked to form the field of fractions of any integral domain. Although ordinary division is not defined on Z, it does possess an important property called the division algorithm: that is, given two integers a and b with b0, there exist unique integers q and r such that a = q b + r and 0 r < | b |, where | b | denotes the absolute value of b. The integer q is called the quotient and r is called the remainder, resulting from division of a by b. This is the basis for the Euclidean algorithm for computing greatest common divisors. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.

8

Order-theoretic propertiesZ is a totally ordered set without upper or lower bound. The ordering of Z is given by: ... 3 < 2 < 1 < 0 < 1 < 2 < 3 < ... An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: 1. if a < b and c < d, then a + c < b + d 2. if a < b and 0 < c, then ac < bc. It follows that Z together with the above ordering is an ordered ring. The integers are the only integral domain whose positive elements are well-ordered, and in which order is preserved by addition.

Integer

9

ConstructionThe integers can be formally constructed as the equivalence classes of ordered pairs of natural numbers (a, b). The intuition is that (a, b) stands for the result of subtracting b from a. To confirm our expectation that 1 2 and 4 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule:

alt=Representation of equivalence classes for the numbers -5 to 5 |Red Points represent ordered pairs of natural numbers. Linked red points are equivalence classes representing the blue integers at the end of the line. |upright=2

precisely when

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; denoting by [(a,b)] the equivalence class having (a,b) as a member, one has:

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:

Hence subtraction can be defined as the addition of the additive inverse:

The standard ordering on the integers is given by: iff It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes. Every equivalence class has a unique member that is of the form (n,0) or (0,n) (or both at once). The natural number n is identified with the class [(n,0)] (in other words the natural numbers are embedded into the integers by map sending n to [(n,0)]), and the class [(0,n)] is denoted n (this covers all remaining classes, and gives the class [(0,0)] a second time since 0=0. Thus, [(a,b)] is denoted by

If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity. This notation recovers the familiar representation of the integers as {... 3,2,1, 0, 1, 2, 3, ...}. Some examples are:

Integer

10

Integers in computingAn integer is often a primitive datatype in computer languages. However, integer datatypes can only represent a subset of all integers, since practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation datatypes (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.). Variable-length representations of integers, such as bignums, can store any integer that fits in the computer's memory. Other integer datatypes are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).

CardinalityThe cardinality of the set of integers is equal to If N = {0, 1, 2, ...} then consider the function: (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from Z to N.

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,0) (1,1) (2,3) (3,5) ... } If N = {1,2,3,...} then consider the function:

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,1) (1,3) (2,5) (3,7) ... } If the domain is restricted to Z then each and every member of Z has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality.

Integer

11

Notes[1] Evans, Nick (1995). "A-Quantifiers and Scope" (http:/ / books. google. com/ ?id=NlQL97qBSZkC). In Bach, Emmon W. Quantification in Natural Languages. Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. pp.262. ISBN0792333527. [2] Miller, Jeff (2010-08-29). "Earliest Uses of Symbols of Number Theory" (http:/ / jeff560. tripod. com/ nth. html). . Retrieved 2010-09-20.

References Bell, E. T., Men of Mathematics. New York: Simon and Schuster, 1986. (Hardcover; ISBN 0-671-46400-0)/(Paperback; ISBN 0-671-62818-6) Herstein, I. N., Topics in Algebra, Wiley; 2 edition (June 20, 1975), ISBN 0-471-01090-1. Mac Lane, Saunders, and Garrett Birkhoff; Algebra, American Mathematical Society; 3rd edition (April 1999). ISBN 0-8218-1646-2. Weisstein, Eric W., " Integer (http://mathworld.wolfram.com/Integer.html)" from MathWorld.

External links The Positive Integers - divisor tables and numeral representation tools (http://www.positiveintegers.org) On-Line Encyclopedia of Integer Sequences (http://www.research.att.com/~njas/sequences/) cf OEIS This article incorporates material from Integer on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Abundant numberIn number theory, an abundant number or excessive number is a number n for which (n) > 2n. Here (n) is the sum-of-divisors function: the sum of all positive divisors of n, including n itself. The value (n)2n is called the abundance of n. An equivalent definition is that the proper divisors of the number (the divisors except the number itself) sum to more than the number. The first few abundant numbers (sequence A005101 [1] in OEIS) are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, As an example, consider the number 24. Its divisors are 1,2, 3, 4, 6, 8, 12 and 24, whose sum is 60. Because 60 is more than 224, the number 24 is abundant. Its abundance is 60224=12. The smallest abundant number not divisible by two, i.e. odd, is 945, and the smallest not divisible by 2 or by 3 is 5391411025 whose prime factors are 52, 7, 11, 13, 17, 19, 23, and 29. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes.[2] If represents the smallest abundant number not divisible by the first k primes then for all we have: for k sufficiently large. Infinitely many even and odd abundant numbers exist. Marc Delglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480.[3] Every proper multiple of a perfect number, and every multiple of an abundant number, is abundant. Also, every integer greater than 20161 can be written as the sum of two abundant numbers.[4] An abundant number which is not a semiperfect number is called a weird number; an abundant number with abundance 1 is called a quasiperfect number. Closely related to abundant numbers are perfect numbers with (n)=2n, and deficient numbers with (n) stop if stop < start: start, stop = stop, start if start < 1: start = 1 if stop < 1: stop = 1 t = [[1]] two-dimensional array c = 1 while c = start: yield t[-1][0] previous row row = [t[-1][-1]] for b in t[-1]: row.append(row[-1] + b) c += 1 number t.append(row) for b in bell_numbers(1, 300): print b The number in the nth row and kth column is the number of partitions of {1, ..., n} such that n is not together in one class with any of the elements k, k+1, ..., n1. For example, there are 7 partitions of {1, ..., 4} such that 4 is not together in one class with either of the elements 2, 3, and there are 10 partitions of {1, ..., 4} such that 4 is not together in one class with element 3. The difference is due to 3 partitions of {1, ..., 4} such that 4 is together in one class with element 2, but not with element 3. This corresponds to the fact that there are 3 partitions of {1, ..., 3} such that 3 is not together in one class with element 2: for counting partitions two elements which are always in one class can be treated as just one element. The 3 appears in the previous row of the table. ## Initialize the triangle as a ## Bell numbers count

18

## Yield the Bell number of the ## Initialize a new row ## Populate the new row ## We have found another Bell ## Append the row to the triangle

Prime Bell numbersThe first few Bell numbers that are primes are: 2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 corresponding to the indices 2, 3, 7, 13, 42 and 55 (sequence A051130 [3] in OEIS). The next prime is B2841, which is approximately 9.30740105 106538. [4] As of 2006, it is the largest known prime Bell number. Phil Carmody showed it was a probable prime in 2002. After 17 months of computation with Marcel Martin's ECPP program Primo, Ignacio Larrosa Caestro proved it to be prime in 2004. He ruled out any other possible primes below B6000, later extended to B30447 by Eric Weisstein.

Bell number

19

References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000110 [2] Spivey, Michael (2008), "A Generalized Recurrence for Bell Numbers" (http:/ / www. cs. uwaterloo. ca/ journals/ JIS/ VOL11/ Spivey/ spivey25. pdf), Journal of Integer Sequences 11, [3] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa051130 [4] http:/ / primes. utm. edu/ primes/ page. php?id=68825

Gian-Carlo Rota (1964). "The Number of Partitions of a Set". American Mathematical Monthly 71 (5): 498504. doi:10.2307/2312585. MR0161805. Lovsz, L. (1993). Combinatorial Problems and Exercises (2nd ed. ed.). Amsterdam, Netherlands: North-Holland. Berend, D.; Tassa, T. (2010). "Improved Bounds on Bell Numbers and on Moments of Sums of Random Variables". Probability and Mathematical Statistics (http://www.math.uni.wroc.pl/~pms/index.php) 30 (2): 185205.

External links Robert Dickau. "Diagrams of Bell numbers" (http://mathforum.org/advanced/robertd/bell.html). Pat Ballew. "Bell numbers" (http://www.pballew.net/Bellno.html). Weisstein, Eric W., " Bell Number (http://mathworld.wolfram.com/BellNumber.html)" from MathWorld. Wagstaff, Samuel S. (1996). "Aurifeuillian factorizations and the period of the Bell numbers modulo a prime" (http://homes.cerias.purdue.edu/~ssw/bell/bell.ps). Mathematics of computation 65 (213): 383391. doi:10.1090/S0025-5718-96-00683-7. MR1325876 Bibcode:1996MaCom..65..383W. Gottfried Helms. "Further properties & Generalization of Bell-Numbers" (http://go.helms-net.de/math/ binomial/04_5_SummingBellStirling.pdf).

Binomial coefficientIn mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written , and it is the coefficient of the xk term in the polynomial expansion of the binomial power (1+x)n. Arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascal's triangle. This family of numbers also arises in many other areas than The binomial coefficients can be arranged to form algebra, notably in combinatorics. For any set containing n Pascal's triangle. elements, the number of distinct k-element subsets of it that can be formed (the k-combinations of its elements) is given by the binomial coefficient . Therefore is often read as "n choose k". The properties of binomial coefficients have led to extending the meaning of the symbol beyond the basic case where n and k are nonnegative integers with k

n; such expressions are then still called binomial coefficients. The notation was introduced by Andreas von Ettingshausen in 1826,[1] although the numbers were already

known centuries before that (see Pascal's triangle). The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, due to Halayudha, on an ancient Hindu classic, Pingala's chandastra. In about 1150, the Hindu mathematician Bhaskaracharya gave a very clear exposition of binomial coefficients in his book

Binomial coefficient Lilavati.[2] Alternative notations include C(n, k), nCk, nCk, , ,[3] in all of which the C stands for combinations or choices.

20

Definition and interpretationsFor natural numbers (taken to include 0) n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of (1 + X)n. The same coefficient also occurs (if k n) in the binomial formula

(valid for any elements x,y of a commutative ring), which explains the name "binomial coefficient". Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that k objects can be chosen from among n objects; more formally, the number of k-element subsets (or k-combinations) of an n-element set. This number can be seen to be equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power (1 + X)n one temporarily labels the term X with an index i (running from 1 to n), then each subset of k indices gives after expansion a contribution Xk, and the coefficient of that monomial in the result will be the number of such subsets. This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n bits (digits 0 or 1) whose sum is k is given by , while the number of ways to write where every ai is a nonnegative integer is given by interpretations are easily seen to be equivalent to counting k-combinations. . Most of these

Computing the value of binomial coefficientsSeveral methods exist to compute the value of k-combinations. without actually expanding a binomial power or counting

Recursive formulaOne has a recursive formula for binomial coefficients

with initial values

The formula follows either from tracing the contributions to Xk in (1 + X)n1(1 + X), or by counting k-combinations of {1, 2, ..., n} that contain n and that do not contain n separately. It follows easily that when k>n, and for all n, so the recursion can stop when reaching such cases. This recursive formula then allows the construction of Pascal's triangle.

Binomial coefficient

21

Multiplicative formulaA more efficient method to compute individual binomial coefficients is given by the formula

This formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, from a set of n objects. The denominator counts the number of distinct sequences that define the same k-combination when order is disregarded.

Factorial formulaFinally there is a formula using factorials that is easy to remember:

where n! denotes the factorial of n. This formula follows from the multiplicative formula above by multiplying numerator and denominator by (n k)!; as a consequence it involves many factors common to numerator and denominator. It is less practical for explicit computation unless common factors are first canceled (in particular since factorial values grow very rapidly). The formula does exhibit a symmetry that is less evident from the multiplicative formula (though it is from the definitions)

Generalization and connection to the binomial seriesThe multiplicative formula allows the definition of binomial coefficients to be extended[4] by replacing n by an arbitrary number (negative, real, complex) or even an element of any commutative ring in which all positive integers are invertible:

With this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling the binomial coefficients:

This formula is valid for all complex numbers and X with |X|n are zero, and the infinite series becomes a finite sum, thereby recovering the binomial formula. However for other values of , including negative integers and rational numbers, the series is really infinite.

Binomial coefficient

22

Pascal's trianglePascal's rule is the important recurrence relation

which can be used to prove by mathematical induction that

is a natural number for all n and k, (equivalent to the

statement that k! divides the product of k consecutive integers), a fact that is not immediately obvious from formula (1). Pascal's rule also gives rise to Pascal's triangle:0: 1: 2: 3: 4: 5: 6: 7: 8: 1 1 8 1 7 28 1 6 21 56 1 5 15 35 70 1 4 10 20 35 56 1 3 6 10 15 21 28 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 1

Row number n contains the numbers

for k = 0,,n. It is constructed by starting with ones at the outside and

then always adding two adjacent numbers and writing the sum directly underneath. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that (x + y)5 = 1 x5 + 5 x4y + 10 x3y2 + 10 x2y3 + 5 x y4 + 1 y5. The differences between elements on other diagonals are the elements in the previous diagonal, as a consequence of the recurrence relation (3) above.

Combinatorics and statisticsBinomial coefficients are of importance in combinatorics, because they provide ready formulas for certain frequent counting problems: There are There are There are There are ways to choose k elements from a set of n elements. See Combination. ways to choose k elements from a set of n if repetitions are allowed. See Multiset. strings containing k ones and n zeros. strings consisting of k ones and n zeros such that no two ones are adjacent.

The Catalan numbers are The binomial distribution in statistics is The formula for a Bzier curve.

Binomial coefficient

23

Binomial coefficients as polynomialsFor any nonnegative integer k, the expression can be simplified and defined as a polynomial divided by k!:

This presents a polynomial in t with rational coefficients. As such, it can be evaluated at any real or complex number t to define binomial coefficients with such first arguments. These "generalized binomial coefficients" appear in Newton's generalized binomial theorem. For each k, the polynomial can be characterized as the unique degree k polynomial p(t) satisfying p(0) = p(1) =

... = p(k 1) = 0 and p(k) = 1. Its coefficients are expressible in terms of Stirling numbers of the first kind, by definition of the latter:

The derivative of

can be calculated by logarithmic differentiation:

Binomial coefficients as a basis for the space of polynomialsOver any field containing Q, each polynomial p(t) of degree at most d is uniquely expressible as a linear combination . The coefficient ak is the kth difference of the sequence p(0), p(1), , p(k). Explicitly,[5]

Integer-valued polynomialsEach polynomial is integer-valued: it takes integer values at integer inputs. (One way to prove this is by induction on k, using Pascal's identity.) Therefore any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, (3.5) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring R of a characteristic 0 field K, a polynomial in K[t] takes values in R at all integers if and only if it is an R-linear combination of binomial coefficient polynomials.

ExampleThe integer-valued polynomial 3t(3t+1)/2 can be rewritten as

Identities involving binomial coefficientsFor any nonnegative integers n and k, This follows from (2) by using (1+x)n = xn(1+x1)n. It is reflected in the symmetry of Pascal's triangle. A combinatorial interpretation of this formula is as follows: when forming a subset of elements (from a set of size ), it is equivalent to consider the number of ways you can pick elements and the number of ways you can

Binomial coefficient exclude elements.

24

The factorial definition lets one relate nearby binomial coefficients. For instance, if k is a positive integer and n is arbitrary, then

and, with a little more work,

Powers of -1A special binomial coefficient is ; it equals powers of -1:

Series involving binomial coefficientsThe formula

is obtained from (2) using x = 1. This is equivalent to saying that the elements in one row of Pascal's triangle always add up to two raised to an integer power. A combinatorial interpretation of this fact involving double counting is given by counting subsets of size 0, size 1, size 2, and so on up to size n of a set S of n elements. Since we count the number of subsets of size i for 0 i n, this sum must be equal to the number of subsets of S, which is known to be 2n. That is, Equation5 is a statement that the power set for a finite set with n elements has size 2n. The formulas

and

follow from (2), after differentiating with respect to x (twice in the latter) and then substituting x = 1. The Chu-Vandermonde identity, which holds for any complex-values m and n and any non-negative integer k, is

and can be found by examination of the coefficient of

in the expansion of (1+x)m(1+x)nm = (1+x)n using

equation (2). When m=1, equation (7a) reduces to equation (3). A similar looking formula, which applies for any integers j, k, and n satisfying 0jkn, is

and

can

be

found

by

examination using

of

the

coefficient

of

in

the

expansion

of

When j=k, equation (7b) gives

Binomial coefficient

25

From expansion (7a) using n=2m, k = m, and (4), one finds

Let F(n) denote the nth Fibonacci number. We obtain a formula about the diagonals of Pascal's triangle

This can be proved by induction using (3) or by Zeckendorf's representation (Just note that the lhs gives the number of subsets of {F(2),...,F(n)} without consecutive members, which also form all the numbers below F(n+1)). Also using (3) and induction, one can show that

Again by (3) and induction, one can show that for k = 0, ... , n1

as well as

which is itself a special case of the result from the theory of finite differences that for any polynomial P(x) of degree less than n,

Differentiating (2) k times and setting x = 1 yields this for and the general case follows by taking linear combinations of these. When P(x) is of degree less than or equal to n,

, when 0 k < n,

where

is the coefficient of degree n in P(x).

More generally for 13b,

where m and d are complex numbers. This follows immediately applying (13b) to the polynomial Q(x):=P(m + dx) instead of P(x), and observing that Q(x) has still degree less than or equal to n, and that its coefficient of degree n is dnan. The infinite series

Binomial coefficient is convergent for k 2. This formula is used in the analysis of the German tank problem. It is equivalent to the formula for the finite sum

26

which is proved for M>m by induction on M. Using (8) one can derive

and .

Identities with combinatorial proofsMany identities involving binomial coefficients can be proved by combinatorial means. For example, the following identity for nonnegative integers (which reduces to (6) when ):

can be given a double counting proof as follows. The left side counts the number of ways of selecting a subset of of at least q elements, and marking q elements among those selected. The right side counts the same parameter, because there are ways of choosing a set of q marks and they occur in all subsets that additionally contain some

subset of the remaining elements, of which there are The recursion formula

where both sides count the number of k-element subsets of {1, 2, . . . , n} with the right hand side rst grouping them into those which contain element n and those which do not. The identity (8) also has a combinatorial proof. The identity reads

Suppose you have

empty squares arranged in a row and you want to mark (select) n of them. There are

ways to do this. On the other hand, you may select your n squares by selecting k squares from among the first n and squares from the remaining n squares. This gives

Now apply (4) to get the result.

Binomial coefficient Sum of coefficients row The number of k-combinations for all k, , is the sum of the nth row (counting from 0) of the

27

binomial coefficients. These combinations are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to , where each digit position is an item from the set of n.

Continuous identitiesCertain trigonometric integrals have values expressible in terms of binomial coefficients: For and

These can be proved by using Euler's formula to convert trigonometric functions to complex exponentials, expanding using the binomial theorem, and integrating term by term.

Generating functionsOrdinary generating functionsFor a fixed n, the ordinary generating function of the sequence is:

For a fixed k, the ordinary generating function of the sequence

is:

The bivariate generating function of the binomial coefficients is:

Another bivariate generating function of the binomial coefficients, which is symmetric, is:

Binomial coefficient

28

Exponential generating functionThe exponential bivariate generating function of the binomial coefficients is:

Divisibility propertiesIn 1852, Kummer proved that if m and n are nonnegative integers and p is a prime number, then the largest power of p dividing equals pc, where c is the number of carries when m and n are added in base p. Equivalently, the exponent of a prime p in equals the number of nonnegative integers j such that the fractional part of k/pj is greater than the fractional part of n/pj. It can be deduced from this that is divisible by n/gcd(n,k). A somewhat surprising result by David Singmaster (1974) is that any integer divides almost all binomial coefficients. More precisely, fix an integer d and let f(N) denote the number of binomial coefficients with n < N such that d divides . Then

Since the number of binomial coefficients

with n < N is N(N+1) / 2, this implies that the density of binomial

coefficients divisible by d goes to 1. Another fact: An integer n2 is prime if and only if all the intermediate binomial coefficients

are divisible by n. Proof: When p is prime, p divides for all 0 1) and (n not in s): s.add(n) n = sum(SQUARE[d] for d in str(n)) return (n == 1)

References[1] [2] [3] [4] [5] [6] "Sad Number" (http:/ / mathworld. wolfram. com/ SadNumber. html). Wolfram Research, Inc.. . Retrieved 2009-09-16. http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa007770 http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa035497 The Prime Database: 10^150006+7426247*10^75000+1 (http:/ / primes. utm. edu/ primes/ page. php?id=76550) Prime Pages entry for 242643801 - 1 (http:/ / primes. utm. edu/ primes/ page. php?id=88847) A161872 (http:/ / en. wikipedia. org/ wiki/ Oeis:a161872)

Additional resources Walter Schneider, Mathews: Happy Numbers (http://web.archive.org/web/20060204094653/http://www. wschnei.de/digit-related-numbers/happy-numbers.html). Weisstein, Eric W., " Happy Number (http://mathworld.wolfram.com/HappyNumber.html)" from MathWorld. Happy Numbers (http://mathforum.org/library/drmath/view/55856.html) at The Math Forum. Guy, Richard (2004). Unsolved Problems in Number Theory (third edition). Springer-Verlag. ISBN 0-387-20860-7.

External links Reg Allenby page (http://www.maths.leeds.ac.uk/pure/staff/allenby/allenby.html)

Highly totient number

96

Highly totient numberA highly totient number k is an integer that has more solutions to the equation (x) = k, where is Euler's totient function, than any integer below it. The first few highly totient numbers are 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440 (sequence A097942 [1] in OEIS). with 1, 3, 4, 5, 6, 10, 11, 17, 21, 31, 34, 37, 38, 49, 54, and 72 totient solutions respectively. The sequence of highly totient numbers is a subset of the sequence of smallest number k with exactly n solutions to (x) = k. These numbers have more ways of being expressed as products of numbers of the form p - 1 and their products than smaller integers. The concept is somewhat analogous to that of highly composite numbers, and in the same way that 1 is the only odd highly composite number, it is also the only odd highly totient number (indeed, the only odd number to not be a nontotient). And just as there are infinitely many highly composite numbers, there are also infinitely many highly totient numbers, though the highly totient numbers get tougher to find the higher one goes, since calculating the totient function involves factorization into primes, something that becomes extremely difficult as the numbers get larger.

References L. Havelock, A Few Observations on Totient and Cototient Valence [2] from PlanetMath

References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa097942 [2] http:/ / aux. planetmath. org/ files/ papers/ 335/ C:TempObsTotientCototientValence. pdf

Highly composite number

97

Highly composite numberA highly composite number (HCN) is a positive integer with more divisors than any positive integer smaller than itself. The initial or smallest twenty-one highly composite numbers are listed in the table at right.number of divisors of 1 2 4 6 12 24 36 48 60 120 180 240 360 720 840 1,260 1,680 2,520 5,040 7,560 10,080 1 2 3 4 6 8 9 10 12 16 18 20 24 30 32 36 40 48 60 64 72

The sequence of highly composite numbers (sequence A002182 [1] in OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 [2] in OEIS). There are an infinite number of highly composite numbers. To prove this fact, suppose that n is an arbitrary highly composite number. Then 2n has more divisors than n (2n itself is a divisor and so are all the divisors of n) and so some number larger than n (and not larger than 2n) must be highly composite as well. Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, but not too many of the same. If we decompose a number n in prime factors like this:

where n is exactly

are prime, and the exponents

are positive integers, then the number of divisors of

Hence, for n to be a highly composite number,

Highly composite number the k given prime numbers pi must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors (for instance 10= 25 may be replaced with 6 = 2 3; both have 4 divisors); the sequence of exponents must be non-increasing, that is ; otherwise, by exchanging two exponents we would again get a smaller number than n with the same number of divisors (for instance 18=2132 may be replaced with 12=2231; both have six divisors). Also, except in two special cases n=4 and n=36, the last exponent ck must equal1. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials. Because the prime factorization of a highly composite number uses all of the first k primes, every highly composite number must be a practical number.[3] Highly composite numbers higher than 6 are also abundant numbers. One need only look at the three or four highest divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also Harshad numbers in base 10. The first HCN that is not a Harshad number is 245,044,800, which has a digit sum of 27, but 27 does not divide evenly into 245,044,800. Many of these numbers are used in traditional systems of measurement, and tend to be used in engineering designs, due to their ease of use in calculations involving vulgar fractions. If Q(x) denotes the number of highly composite numbers which are less than or equal to x, then there exist two constants a and b, both bigger than 1, so that (ln x)a Q(x) (ln x)b. with the first part of the inequality proved by Paul Erds in 1944 and the second part by Jean-Louis Nicolas in 1988.

98

ExamplesThe highly composite number : 10,080.10,080 = (2 2 2 2 2) (3 3) 5 7 By (2) above, 10,080 has exactly seventy-two divisors. 1 10,080 7 1,440 15 672 28 360 42 240 70 144 2 5,040 8 1,260 16 630 30 336 45 224 72 140 3 3,360 9 1,120 18 560 32 315 48 210 80 126 4 2,520 10 1,008 20 504 35 288 56 180 84 120 5 2,016 12 840 21 480 36 280 60 168 90 112 6 1,680 14 720 24 420 40 252 63 160 96 105

Highly composite number

99

Note: The bolded numbers are themselves highly composite numbers. Only the twentieth highly composite number 7560 (=32520) is absent. 10080 is a so-called 7-smooth number, (sequence A002473 [4] in OEIS).

The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes: , where is the sequence of successive prime numbers, and all omitted terms (a22 to a228) are factors with exponent equal to one (i.e. the number is ). [5]

Prime factor subsetsFor any highly composite number, if one takes any subset of prime factors for that number and their exponents, the resulting number will have more divisors than any smaller number that uses the same prime factors. For example for the highly composite number 720 which is 24325 we can be sure that 144 which is 2432 has more divisors than any smaller number that has only the prime factors 2 and 3 80 which is 245 has more divisors than any smaller number that has only the prime factors 2 and 5 45 which is 325 has more divisors than any smaller number that has only the prime factors 3 and 5 If this were untrue for any particular highly composite number and subset of prime factors, we could exchange that subset of primefactors and exponents for the smaller number using the same primefactors and get a smaller number with at least as many divisors. This property is useful for finding highly composite numbers.

References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa002182 [2] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa005179 [3] Srinivasan, A. K. (1948), "Practical numbers" (http:/ / www. ias. ac. in/ jarch/ currsci/ 17/ 179. pdf), Current Science 17: 179180, MR0027799, . [4] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa002473 [5] Flammenkamp, Achim, Highly Composite Numbers (http:/ / wwwhomes. uni-bielefeld. de/ achim/ highly. html), .

External links Weisstein, Eric W., " Highly Composite Number (http://mathworld.wolfram.com/HighlyCompositeNumber. html)" from MathWorld. Earth360: Versatile Numbers: Self-Organization, Emergence, and Economics (http://earth360.com/ math-versatile.html) Algorithm for computing Highly Composite Numbers (http://web.archive.org/web/19980707133810/www. math.princeton.edu/~kkedlaya/math/hcn-algorithm.tex) First 10000 Highly Composite Numbers (http://web.archive.org/web/19980707133953/www.math. princeton.edu/~kkedlaya/math/hcn10000.txt.gz) Achim Flammenkamp, First 779674 HCN with sigma,tau,factors (http://wwwhomes.uni-bielefeld.de/achim/ highly.html)

Home prime

100

Home primeIn number theory, the home prime HP(n) of an integer n greater than 1 is the prime obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth intermediate stage in the process of determining HP(n) is designated HPn(m). For instance, HP(10)=773, as 10 factors as 25 yielding HP10(1)=25, 25 factors as 55 yielding HP10(2)=HP25(1)=55, 55=511 implies HP10(3)=HP25(2)=HP55(1)=511, and 511=773 gives HP10(4)=HP25(3)=HP55(2)=HP511(1)=773, a prime number. Some sources use the alternative notation HPn for the homeprime, leaving out parentheses. Investigations into home primes make up a minor side issue in number theory. Its questions have served as test fields for the implementation of efficient algorithms for factoring composite numbers, but the subject is really one in recreational mathematics. The outstanding computational problem at present (January, 2011) is whether HP(49)=HP(77) can be calculated in practice. As each iteration is greater than the previous up until a prime is reached, factorizations generally grow more difficult so long as an end is not reached. The pursuit of HP(49) is now a process of factoring a 210-digit composite factor of HP49(106) after a break was achieved on 8 February 2010 with the calculation of HP49(100) after work lasting the larger part of a decade and an extensive use of computational resources and the successful factorization of a 178-digit composite in HP49(104) into 88- and 90-digit primes on 11 January 2011. Details of the history of this search, as well as the sequences leading to home primes for all other numbers through 100, are maintained at Patrick De Geest's worldofnumbers website. A wiki primarily associated with the Great Internet Mersenne Prime Search maintains the complete known data through 1000 in base 10 and also has lists for the bases 2 through9. Aside from the computational problems that have had so much time devoted to them, it appears absolute proof of existence of a home prime for any specific number might entail its effective computation. In purely heuristic terms, the existence has probability 1 for all numbers, but such heuristics make assumptions about numbers drawn from a wide variety of processes that, though they likely are correct, fall short of the standard of proof usually required of mathematical claims.

Early history and additional terminologyWhile it's unlikely that the idea was not conceived of numerous times in the past, the first reference in print appears to be an article written in 1990 in a small and now-defunct publication called Recreational and Educational Computation. The same person who authored that article, Jeffrey Heleen, revisited the subject in the 19967 volume of the Journal of Recreational Mathematics in an article entitled Family Numbers: Constructing Primes By Prime Factor Splicing, which included all of the results HP(n) for n through 100 other than the ones still unresolved. It also included a now-obsolete list of 3-digit unresolved numbers (The 58 listed have been cut precisely in half as of January, 2011). It appears that this article is largely responsible for provoking attempts by others to resolve the case involving 49 and77. The article uses the terms daughter and parent to describe composites and the primes that they lead to, with numbers leading to the same home prime called siblings (even if one is an iterate of another), and calls the number of iterations required to reach a parent, the persistence of a number under the map to obtain a home prime, the number of lives. The brief article does little other than state the origins of the subject, define terms, give a couple of examples, mention machinery and methods used at the time, and then provide tables. It appears that Mr. De Geest is responsible for the notation now in use. The OEIS also uses homeliness as the term for the number of numbers, including the prime itself, that have a certain prime as its home prime.

Home prime

101

References1. http://oeis.org/A037274 2. http://www.worldofnumbers.com/topic1.htm 3. http://mathworld.wolfram.com/HomePrime.html 4. http://www.mersennewiki.org/index.php/Home_Primes_Search 5. J. Heleen, Family Numbers: Constructing Primes By Prime Factor Splicing, J. Rec. Math., 28, pp.1169, 1996-7 6. J. Heleen, Family Numbers: Mathemagical Black Holes, Recreational and Educational Computing, 5:5, p.6, 1990

Hyperperfect numberIn mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k((n) n 1) holds, where (n) is the divisor function (i.e., the sum of all positive divisors of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect. The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, ... (sequence A034897 [1] in OEIS), with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... (sequence A034898 [2] in OEIS). The first few k-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... (sequence A007592 [3] in OEIS).

List of hyperperfect numbersThe following table lists the first few k-hyperperfect numbers for some values of k, together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of k-hyperperfect numbers:k 1 2 3 4 6 10 11 12 18 19 30 31 35 40 48 59 60 A028500 A028501 [7] [8] A028499 [6] OEIS A000396 A007593 [4] [5] Some known k-hyperperfect numbers 6, 28, 496, 8128, 33550336, ... 21, 2133, 19521, 176661, 129127041, ... 325, ... 1950625, 1220640625, ... 301, 16513, 60110701, 1977225901, ... 159841, ... 10693, ... 697, 2041, 1570153, 62722153, 10604156641, 13544168521, ... 1333, 1909, 2469601, 893748277, ... 51301, ... 3901, 28600321, ... 214273, ... 306181, ... 115788961, ... 26977, 9560844577, ... 1433701, ... 24601, ...

Hyperperfect number

10266 75 78 91 100 108 126 132 136 138 140 168 174 180 190 192 198 206 222 228 252 276 282 296 342 348 350 360 366 372 396 402 408 414 430 438 480 522 546 296341, ... 2924101, ... 486877, ... 5199013, ... 10509080401, ... 275833, ... 12161963773, ... 96361, 130153, 495529, ... 156276648817, ... 46727970517, 51886178401, ... 1118457481, ... 250321, ... 7744461466717, ... 12211188308281, ... 1167773821, ... 163201, 137008036993, ... 1564317613, ... 626946794653, 54114833564509, ... 348231627849277, ... 391854937, 102744892633, 3710434289467, ... 389593, 1218260233, ... 72315968283289, ... 8898807853477, ... 444574821937, ... 542413, 26199602893, ... 66239465233897, ... 140460782701, ... 23911458481, ... 808861, ... 2469439417, ... 8432772615433, ... 8942902453, 813535908179653, ... 1238906223697, ... 8062678298557, ... 124528653669661, ... 6287557453, ... 1324790832961, ... 723378252872773, 106049331638192773, ... 211125067071829, ...

Hyperperfect number

103570 660 672 684 774 810 814 816 820 968 972 978 1050 1410 2772 3918 9222 9828 14280 23730 31752 55848 67782 92568 100932 A034916 A028502 [9] 1345711391461, 5810517340434661, ... 13786783637881, ... 142718568339485377, ... 154643791177, ... 8695993590900027, ... 5646270598021, ... 31571188513, ... 31571188513, ... 1119337766869561, ... 52335185632753, ... 289085338292617, ... 60246544949557, ... 64169172901, ... 80293806421, ... 95295817, 124035913, ... 61442077, 217033693, 12059549149, 60174845917, ... 404458477, 3426618541, 8983131757, 13027827181, ... 432373033, 2797540201, 3777981481, 13197765673, ... 848374801, 2324355601, 4390957201, 16498569361, ... 2288948341, 3102982261, 6861054901, 30897836341, ... [10] 4660241041, 7220722321, 12994506001, 52929885457, 60771359377, ... 15166641361, 44783952721, 67623550801, ... 18407557741, 18444431149, 34939858669, ... 50611924273, 64781493169, 84213367729, ... 50969246953, 53192980777, 82145123113, ...

It can be shown that if k > 1 is an odd integer and p = (3k + 1) / 2 and q = 3k + 4 are prime numbers, then pq is k-hyperperfect; Judson S. McCranie has conjectured in 2000 that all k-hyperperfect numbers for odd k > 1 are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if p q are odd primes and k is an integer such that k(p + q) = pq - 1, then pq is k-hyperperfect. It is also possible to show that if k > 0 and p = k + 1 is prime, then for all i > 1 such that q = pi p + 1 is prime, n = pi 1 q is k-hyperperfect. The following table lists known values of k and corresponding values of i for which n is k-hyperperfect:

Hyperperfect number

104

k 16 22 28 36 42 46 52 58 72 88 96

OEIS A034922

Values of i

[11] 11, 21, 127, 149, 469, ... 17, 61, 445, ... 33, 89, 101, ... 67, 95, 341, ...

A034923 A034924

[12] 4, 6, 42, 64, 65, ... [13] 5, 11, 13, 53, 115, ... 21, 173, ... 11, 117, ... 21, 49, ...

A034925

[14] 9, 41, 51, 109, 483, ... 6, 11, 34, ...

100 A034926 [15] 3, 7, 9, 19, 29, 99, 145, ...

HyperdeficiencyThe newly-introduced mathematical concept of hyperdeficiency is related to the hyperperfect numbers. Definition (Minoli 2010): For any integer n and for integer k, - 0.

Further readingArticles Daniel Minoli, Robert Bear (Fall 1975), "Hyperperfect numbers", Pi Mu Epsilon Journal 6 (3): 153157. Daniel Minoli (Dec 1978), "Sufficient forms for generalized perfect numbers", Annales de la Facult des Sciences UNAZA 4 (2): 277302. Daniel Minoli (Feb. 1981), "Structural issues for hyperperfect numbers", Fibonacci Quarterly 19 (1): 614. Daniel Minoli (April 1980), "Issues in non-linear hyperperfect numbers", Mathematics of Computation 34 (150): 639645. Daniel Minoli (October 1980), "New results for hyperperfect numbers", Abstracts of the American Mathematical Society 1 (6): 561. Daniel Minoli, W. Nakamine (1980), "Mersenne numbers rooted on 3 for number theoretic transforms", International Conference on Acoustics, Speech, and Signal Processing. Judson S. McCranie (2000), "A study of hyperperfect numbers" [16], Journal of Integer Sequences 3.

Hyperperfect number

105

Books Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

External links MathWorld: Hyperperfect number [17] has a long list of hyperperfect numbers under Data [18]

References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034897 [2] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034898 [3] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa007592 [4] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000396 [5] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa007593 [6] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa028499 [7] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa028500 [8] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa028501 [9] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa028502 [10] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034916 [11] [12] [13] [14] [15] [16] [17] [18] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034922 http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034923 http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034924 http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034925 http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa034926 http:/ / www. math. uwaterloo. ca/ JIS/ VOL3/ mccranie. html http:/ / mathworld. wolfram. com/ HyperperfectNumber. html http:/ / j. mccranie. home. comcast. net

Juggler sequence

106

Juggler sequenceIn recreational mathematics a juggler sequence is an integer sequence that starts with a positive integer a0, with each subsequent term in the sequence defined by the recurrence relation:

BackgroundJuggler sequences were publicised by American mathematician and author Clifford A. Pickover.[1] The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler.[2] For example, the juggler sequence starting with a0 = 3 is

If a juggler sequence reaches 1, then all subsequent terms are equal to 1. It is conjectured that all juggler sequences eventually reach 1. This conjecture has been verifed for initial terms up to 106,[3] but has not been proved. Juggler sequences therefore present a problem that is similar to the Collatz conjecture, about which Paul Erds stated that "mathematics is not yet ready for such problems". For a given initial term n we define l(n) to be the number of steps which the juggler sequence starting at n takes to first reach 1, and h(n) to be the maximum value in the juggler sequence starting at n. For small values of n we have:n 2 3 4 5 6 7 8 9 Juggler sequence 2, 1 3, 5, 11, 36, 6, 2, 1 4, 2, 1 5, 11, 36, 6, 2, 1 6, 2, 1 7, 18, 4, 2, 1 8, 2, 1 9, 27, 140, 11, 36, 6, 2, 1 l(n) (sequence A007320 1 6 2 5 2 4 2 7 7 [4] in OEIS) h(n) (sequence A094716 2 36 4 36 6 18 8 140 36 [5] in OEIS)

10 10, 3, 5, 11, 36, 6, 2, 1

Juggler sequences can reach very large values before descending to 1. For example, the juggler sequence starting at a0 = 37 reaches a maximum value of 24906114455136. Harry J. Smith has determined that the juggler sequence starting at a0 = 48443 reaches a maximum value at a60 with 972,463 digits, before reaching 1 at a157.[6]

Juggler sequence

107

References[1] [2] [3] [4] [5] [6] Pickover, Clifford A. (1992). Computers and the Imagination. St. Martin's Press. pp.Chapter 40. ISBN978-0312083434. Pickover, Clifford A. (2002). The Mathematics of Oz. Cambridge University Press. pp.Chapter 45. ISBN978-0521016780. *Weisstein, Eric W., " Juggler Sequence (http:/ / mathworld. wolfram. com/ JugglerSequence. html)" from MathWorld. http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa007320 http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa094716 Letter from Harry J. Smith to Cliiford A. Pickover, 27th June 1992 (http:/ / web. archive. org/ web/ 20091027155431/ http:/ / geocities. com/ hjsmithh/ Juggler/ Juggle3L. html)

External links Juggler sequence calculator (http://members.chello.nl/k.ijntema/juggler.html) at Collatz Conjecture Calculation Center Juggler Number pages (http://web.archive.org/web/20091027103635/http://geocities.com/hjsmithh/ Juggler/index.html) by Harry J. Smith

Kolakoski sequenceIn mathematics, the Kolakoski sequence is an infinite list that begins with 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,... The rules for generating this sequence are as follows: 1) The only numbers allowed are 1 and 2 2) Each number tells us how many numbers to append to the sequence 2a) 1 tells us to append one more number 2b) 2 tells us to append two more numbers 3) We can have no more than two of the same number in sequence 3a) This is because if we write 222 that means that somewhere in the sequence it told us to append 3 twos, but the only numbers allowed are 1 and 2 4) Every time we "read" a new number, we alternate between writing 1 and 2 5) A0=1 This is an example of a self-generating sequence, with the first term as 1. Each individual number describes how many numbers to write next. This is called the run of the numbers. To start the sequence we have to write one 2 after the 1 because the 1 tells us to write one number, and because our initial condition "wrote" the one we must alternate by writing the 2. This gives us as our sequence so far (1 2). Next, The 2 tells us that the length, or the run, of the next set of numbers must be 2, but we cannot have any more than two ones or two twos in sequence because that would mean that the sequence told us to write three 2's, which would mean a 3 would have to be in the sequence, which is not allowed. Therefore, we must write 2 1. This gives us our sequence so far (1 2 2 1). The next number in the list is a 2, so we must write two numbers. The next two numbers that we should write is a 1; however, we cannot write three 1's in a row, so we must append 1 2 to the sequence. This gives us our sequence so far (1 2 2 1 1 2). The fourth number in the list is a 1. As we wrote a 2 last we write a 1 now. This gives us our sequence so far (1 2 2 1 1 2 1). The fifth number in the sequence is a 1. As we wrote a 1 before, we must write a 2 now. This gives us our sequence so far (1 2 2 1 1 2 1 2). This continues on. Another explanation for the generation of the Kolakoski sequence is indicated here: (1) write 1; read it as the number of 1's to write before switching to 2;

Kolakoski sequence (2) write 2; read it as the number of 2's to write before switching back to 1; (3) so far... 1,2,2; read the new 2 as the number of 1's to write; (4) so far... 1,2,2,1,1; read the new 1,1 as the number of 2's and then 1's to write; (5) so far... 1,2,2,1,1,2,1; continue generating forever. To truly understand how the sequence is generated, follow either of the two examples above and write out the first few terms. It seems plausible that the density of 1's is 1/2, but this conjecture remains unproved. This and related simply stated unsolved problems are presented at Integer Sequences and Arrays. [1] Attempts to solve them are referenced at MathWorld [2] and sequence A000002 [3] at On-Line Encyclopedia of Integer Sequences. In the unpublished technical report Notes on the Kolakoski Sequence [4], Chvtal proved that the upper density of 1's is less than 0.50084. Kolakoski's self-generating sequence has attracted the interest of computer scientists as well as mathematicians. For example, in A New Kind of Science, p. 895, Stephen Wolfram describes the Kolakoski sequence in connection with the history of cyclic tag systems.

108

Kolakoski fanThe Kolakoski fan is the following array: 1 2 22 1122 121122 211212211 (and so on, where the initial entries are given by the Kolakoski sequence, and the block lengths in each row are given by the entries in the previous row) Associated with this array, indexed as A143477 [5] (in the On-Line Encyclopedia of Integer Sequences) are many other such arrays, such as A111090 [6], for which it is conjectured that the growth rate of row-length is asymptotic to c(3/2)n for some constant c.

References J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 337. Vaek Chvtal, "Notes on the Kolakoski Sequence", DIMACS Technical Report 93-84, December 1993. F. M. Dekking, "What Is the Long Range Order in the Kolakoski Sequence?" In Proceedings of the NATO Advanced Study Institute held in Waterloo, ON, August 21-September 1, 1995 (dd. R. V. Moody). Dordrecht, Netherlands: Kluwer, pp. 115-125, 1997. M. S. Keane, "Ergodic Theory and Subshifts of Finite Type." In Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces (ed. T. Bedford and M. Keane). Oxford, England: Oxford University Press, pp. 35-70, 1991. William Kolakoski, proposal 5304, American Mathematical Monthly 72 (1965), 674; for a partial solution, see "Self Generating Runs," by Necdet oluk, Amer. Math. Mon. 73 (1966), 681-2. J. C. Lagarias, "Number Theory and Dynamical Systems." In The Unreasonable Effectiveness of Number Theory (ed. S. A. Burr). Providence, RI: Amer. Math. Soc., pp. 35-72, 1992. G. Paun and A. Salomaa, "Self-Reading Sequences." Amer. Math. Monthly 103, 166-168, 1996.

Kolakoski sequence Bertran Steinsky, "A Recursive Formula for the Kolakoski Sequence A000002," Journal of Integer Sequences 9 (2006) Article 06.3.7. J.M. Fedou and G. Fici, "Some remarks on differentiable sequences and recursivity", "Journal of Integer Sequences" 13(3) (2010) Article 10.3.2.

109

External links Kolakoski Sequence [2] Online Encyclopedia of Integer Sequences [7] (search "Kolakoski") Kolakoski Constant to 25000 Digits. [8]

References[1] [2] [3] [4] [5] [6] [7] http:/ / faculty. evansville. edu/ ck6/ integer/ index. html http:/ / mathworld. wolfram. com/ KolakoskiSequence. html http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000002 http:/ / dimacs. rutgers. edu/ TechnicalReports/ abstracts/ 1993/ 93-84. html http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa143477 http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa111090 http:/ / www. research. att. com/ ~njas/ sequences/

[8] http:/ / pi. lacim. uqam. ca/ piDATA/ Kolakoski. txt

Lucky numberIn number theory, a lucky number is a natural number in a set which is generated by a "sieve" similar to the Sieve of Eratosthenes that generates the primes. Begin with a list of integers starting with 1: 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, Every second number (all even numbers) is eliminated, leaving only the odd integers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25,

The second term in this sequence is 3. Every third number which remains in the list is eliminated: 1, 3, 7, 9, 13, 15, 19, 21, 25,

The third surviving number is now 7, so every seventh number that remains is eliminated: 1, 3, 7, 9, 13, 15, 21, 25,

As this procedure is repeated indefinitely, the survivors are the lucky numbers: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, ... (sequence A000959 [1] in OEIS).

Lucky number

110

The term was introduced in 1955 in a paper by Gardiner, Lazarus, Metropolis and Ulam. They suggest also calling its defining sieve the sieve of Josephus Flavius[2] because of its similarity with the counting-out game in the Josephus problem. Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. Because of these apparent connections with the prime numbers, some mathematicians have suggested that these properties may be found in a larger class of sets of numbers generated by sieves of a certain unknown form, although there is little theoretical basis for this conjecture. Twin lucky numbers and twin primes also appear to occur with similar frequency. A lucky prime is a lucky number that is prime. It is not known whether there are infinitely many lucky primes. The first few are 3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193 (sequence A031157 [3] in OEIS).An animation demonstrating the lucky number sieve. The numbers in red are lucky numbers.

References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000959 [2] V. Gardiner, R. Lazarus, N. Metropolis and S. Ulam, "On certain sequences of integers defined by sieves", Mathematics Magazine 29:3 (1955), pp. 117122. [3] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa031157

External links Peterson, Ivars. MathTrek: Martin Gardner's Lucky Number (http://www.sciencenews.org/sn_arc97/9_6_97/ mathland.htm) Weisstein, Eric W., " Lucky Number (http://mathworld.wolfram.com/LuckyNumber.html)" from MathWorld. Lucky Numbers (http://demonstrations.wolfram.com/LuckyNumbers/) by Enrique Zeleny, The Wolfram Demonstrations Project.

Lucas number

111

Lucas numberThe Lucas numbers are an integer sequence named after the mathematician Franois douard Anatole Lucas (18421891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.

DefinitionLike the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, i.e. it is a Fibonacci integer sequence. Consequently, the ratio between two consecutive Lucas numbers converges to the golden ratio. However, the first two Lucas numbers are L0 = 2 and L1 = 1 instead of 0 and 1, and the properties of Lucas numbers are therefore somewhat different from those of Fibonacci numbers. A Lucas number may thus be defined as follows:

The sequence of Lucas numbers begins: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (sequence A000032 [1] in OEIS)

Extension to negative integersUsing Ln-2 = Ln - Ln-1, one can extend the Lucas numbers to negative integers to obtain a doubly infinite sequence : ..., -11, 7, -4, 3, -1, 2, 1, 3, 4, 7, 11, ... (terms for are shown). The formula for terms with negative indices in this sequence is

Relationship to Fibonacci numbersThe Lucas numbers are related to the Fibonacci numbers by the identities Their closed formula is given as: , and thus as approaches +, the ratio approaches

where

is the Golden ratio. Alternatively,

is the closest integer to

.

Lucas number

112

Congruence relationLn is congruent to 1 mod n if n is prime, but some composite values of n also have this property.

Lucas primesA Lucas prime is a Lucas number that is prime. The first few Lucas primes are 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, ... (sequence A005479 [2] in OEIS) If Ln is prime then n is either 0, prime, or a power of 2.[3] L values of . is prime for = 1, 2, 3, and 4 and no other known

Lucas polynomialsThe Lucas polynomials Ln(x) are a polynomial sequence derived from the Lucas numbers in the same way as Fibonacci polynomials are derived from the Fibonacci numbers. Lucas polynomials are defined by the following recurrence relation:

Lucas polynomials can be expressed in terms of Lucas sequences as

The first few Lucas polynomials are:

The Lucas numbers are recovered by evaluating the polynomials at x=1. The degree of Ln(x) is n. The ordinary generating function for the sequence is

References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000032 [2] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa005479 [3] Chris Caldwell, " The Prime Glossary: Lucas prime (http:/ / primes. utm. edu/ glossary/ page. php?sort=LucasPrime)" from The Prime Pages.

External links Weisstein, Eric W., " Lucas Number (http://mathworld.wolfram.com/LucasNumber.html)" from MathWorld. Weisstein, Eric W., " Lucas Polynomial (http://mathworld.wolfram.com/LucasPolynomial.html)" from MathWorld. Dr Ron Knott (http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucasNbs.html) Lucas numbers and the Golden Section (http://milan.milanovic.org/math/english/lucas/lucas.html)

Lucas number A Lucas Number Calculator can be found here. (http://www.plenilune.pwp.blueyonder.co.uk/ fibonacci-calculator.asp) A Tutorial on Generalized Lucas Numbers (http://nakedprogrammer.com/LucasNumbers.aspx)

113

Padovan sequenceThe Padovan sequence is the sequence of integers P(n) defined by the initial values and the recurrence relation

The first few values of P(n) are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... (sequence A000931 [1] in OEIS) The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom. Hans van der Laan : Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996. The above definition is the one given by Ian Stewart and by MathWorld. Other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets.

Spiral of equilateral triangles with side lengths which follow the Padovan sequence.

Recurrence relationsIn the spiral, each triangle shares a side with two others giving a visual proof that the Padovan sequence also satisfies the recurrence relation

Starting from this, the defining recurrence and other recurrences as they are discovered, one can create an infinite number of further recurrences by repeatedly replacing by The Perrin sequence satisfies the same recurrence relations as the Padovan sequence, although it has different initial values. This is a property of recurrence relations. The Perrin sequence can be obtained from the Padovan sequence by the following formula:

Padovan sequence

114

Extension to negative parametersAs with any sequence defined by a recurrence relation, Padovan numers P(m) for m n p(k, n) = 1 if k = n p(k, n) = p(k+1, n) + p(k, n k) otherwise. This function tends to exhibit deceptive behavior. p(1, 4) = 5 p(2, 8) = 7 p(3, 12) = 9 p(4, 16) = 11 p(5, 20) = 13 p(6, 24) = 16 Our original function p(n) is just p(1, n). The values of this function:k 1 n 1 2 3 4 5 6 7 8 9 10 1 2 3 5 7 11 15 22 30 2 0 1 1 2 2 4 4 7 8 3 0 0 1 1 1 2 2 3 4 5 4 0 0 0 1 1 1 1 2 2 3 5 0 0 0 0 1 1 1 1 1 2 6 0 0 0 0 0 1 1 1 1 1 7 0 0 0 0 0 0 1 1 1 1 8 0 0 0 0 0 0 0 1 1 1 9 0 0 0 0 0 0 0 0 1 1 10 0 0 0 0 0 0 0 0 0 1

42 12

Partition number

120

Generating functionA generating function for p(n) is given by the reciprocal of Euler's function:

Expanding each term on the right-hand side as a geometric series, we can rewrite it as (1 + x + x2 + x3 + ...)(1 + x2 + x4 + x6 + ...)(1 + x3 + x6 + x9 + ...) .... The xn term in this product counts the number of ways to write n = a1 + 2a2 + 3a3 + ... = (1 + 1 + ... + 1) + (2 + 2 + ... + 2) + (3 + 3 + ... + 3) + ..., where each number i appears ai times. This is precisely the definition of a partition of n, so our product is the desired generating function. More generally, the generating function for the partitions of n into numbers from a set A can be found by taking only those terms in the product where k is an element of A. This result is due to Euler. The formulation of Euler's generating function is a special case of a q-Pochhammer symbol and is similar to the product formulation of many modular forms, and specifically the Dedekind eta function. It can also be used in conjunction with the pentagonal number theorem to derive a recurrence for the partition function stating that: p(k) = p(k 1) + p(k 2) p(k 5) p(k 7) + p(k 12) + p(k 15) p(k 22) ... where p(0) is taken to equal 1, p(k) is zero for negative k, and the sum is taken over all generalized pentagonal numbers of the form n(3n 1), for n running over positive and negative integers: successively taking n = 1, 1, 2, 2, 3, 3, 4, 4 ..., generates the values 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, .... The signs in the summation continue to alternate+,+,,,+,+,...

Table of valuesSome values of the partition function are as follows (sequence A000041 [2] in OEIS): p(1) = 1 p(2) = 2 p(3) = 3 p(4) = 5 p(5) = 7 p(6) = 11 p(7) = 15 p(8) = 22 p(9) = 30 p(10) = 42 p(100) = 190,569,292 p(200) = 3,972,999,029,388 p(1000) = 24,061,467,864,032,622,473,692,149,727,991 2.41031.

As of February 2010, the largest known prime number of this kind is p(29099391), with 6002 decimal digits, found by Predrag Minovic.[3]

Partition number

121

Asymptotic behaviourAn asymptotic expression for p(n) is given by

This asymptotic formula was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920. Considering p(1000), the asymptotic formula gives about 2.44021031, reasonably close to the exact answer given above. In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan's results by providing a convergent series expression for p(n). It is

where

Here, the notation (m,n)=1 implies that the sum should occur only over the values of m that are relatively prime to n. The function s(m,k) is a Dedekind sum. The proof of Rademacher's formula is interesting in that it involves Ford circles, Farey sequences, modular symmetry and the Dedekind eta function in a central way.

CongruencesSrinivasa Ramanujan is credited with discovering that "congruences" in the number of partitions exist for integers ending in 4 and 9.

For instance, the number of partitions for the integer 4 is 5. For the integer 9, the number of partitions is 30; for 14 there are 135 partitions. He also discovered congruences related to 7 and 11:

Since 5, 7, and 11 are consecutive primes, one might think that there would be such a congruence for the next prime 13, for some a. This is, however, false. It can also be shown that there is no congruence of the form for any prime b other than 5, 7, or 11. In the 1960s, A. O. L. Atkin of the University of Illinois at Chicago discovered additional congruences for small prime moduli. For example:

In 2000, Ken Ono of the University of WisconsinMadison proved that there are such congruences for every prime modulus. A few years later Ono, together with Scott Ahlgren of the University of Illinois, proved that there are partition congruences modulo every integer coprime to 6.[4]

Partition number

122

Restricted partitionsAmong the 22 partitions for the number 8, 6 contain only odd parts: 7+1 5+3 5+1+1+1 3+3+1+1 3+1+1+1+1+1 1+1+1+1+1+1+1+1

If we count the partitions of 8 with distinct parts, we also obtain the number 6: 8 7+1 6+2 5+3 5+2+1 4+3+1

It is true for all positive numbers that the number of partitions with odd parts always equals the number of partitions with distinct parts. This result was proved by Leonard Euler in 1748.[5] Some similar results about restricted partitions can be obtained by the aid of a visual tool, a Ferrers graph (also called Ferrers diagram, since it is not a graph in the graph-theoretical sense, or sometimes Young diagram, alluding to the Young tableau).

Ferrers diagramThe partition 6+4+3+1 of the positive number 14 can be represented by the following diagram; these diagrams are named in honor of Norman Macleod Ferrers:

6+4+3+1

The 14 circles are lined up in 4 columns, each having the size of a part of the partition. The diagrams for the 5 partitions of the number 4 are listed below:

4

= 3+1 =

2+2

= 2+1+1 =

1+1+1+1

If we now flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14:

Partition number

123

6+4+3+1

=

4+3+3+2+1+1

By turning the rows into columns, we obtain the partition 4+3+3+2+1+1 of the number 14. Such partitions are said to be conjugate of one another. In the case of the number 4, partitions 4 and 1+1+1+1 are conjugate pairs, and partitions 3+1 and 2+1+1 are conjugate of each other. Of particular interest is the partition 2+2, which has itself as conjugate. Such a partition is said to be self-conjugate. Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts. Proof (outline): The crucial observation is that every odd part can be "folded" in the middle to form a self-conjugate diagram:

One can then obtain a bijection between the set of partitions with distinct odd parts and the set of self-conjugate partitions, as illustrated by the following example:

9+7+3 Dist. odd

=

5+5+4+3+2 self-conjugate

Similar techniques can be employed to establish, for example, the following equalities: The number of partitions of n into no more than k parts is the same as the number of partitions of n into parts no larger thank. The number of partitions of n into no more than k parts is the same as the number of partitions of n+k into exactly k parts.

Partition number

124

Notes[1] [2] [3] [4] http:/ / www. aimath. org/ news/ partition/ http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa000041 http:/ / primes. utm. edu/ top20/ page. php?id=54 Ono, Ken; Ahlgren, Scott (2001). "Congruence properties for the partition function" (http:/ / www. math. wisc. edu/ ~ono/ reprints/ 061. pdf). Proceedings of the National Academy of Sciences 98 (23): 12,88212,884. doi:10.1073/pnas.191488598. . [5] Andrews, George E. Number Theory. W. B. Saunders Company, Philadelphia, 1971. Dover edition, page 149150.

References George E. Andrews, The Theory of Partitions (1976), Cambridge University Press. ISBN 0-521-63766-X . T