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A000688 Number of Abelian groups of order n; number of factorizations of n into prime powers.
(Formerly M0064 N0020)
64
1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Equivalently, number of Abelian groups with n conjugacy classes. - Michael Somos, Aug 10 2010

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3, 1).

Also number of rings with n elements that are the direct product of fields; these are the commutative rings with n elements having no nilpotents; likewise the commutative rings where for every element x there is a k > 0 such that x^(k+1) = x. - Franklin T. Adams-Watters, Oct 20 2006

Range is A033637.

a(n) = 1 if and only if n is from A005117 (squarefree numbers). See the Ahmed Fares comment there, and the formula for n>=2 below. - Wolfdieter Lang, Sep 09 2012

Also, from a theorem of Molnár (see [Molnár]), the number of (non-isomorphic) abelian groups of order 2*n + 1 is equal to the number of non-congruent lattice Z-tilings of R^n by crosses, where a "cross" is a unit cube in R^n for which at each facet is attached another unit cube (Z, R are the integers and reals, respectively). (Cf. [Horak].) - L. Edson Jeffery, Nov 29 2012

Zeta(k*s) is the Dirichlet generating function of the characteristic function of numbers which are k-th powers (k=1 in A000012, k=2 in A010052, k=3 in A010057, see arXiv:1106.4038 Section 3.1). The infinite product over k (here) is the number of representations n=product_i (b_i)^(e_i) where all exponents e_i are distinct and >=1. Examples: a(n=4)=2: 4^1 = 2^2. a(n=8)=3: 8^1 = 2^1*2^2 = 2^3. a(n=9)=2: 9^1 = 3^2. a(n=12)=2: 12^1 = 3*2^2. a(n=16)=5: 16^1 = 2*2^3 = 4^2 = 2^2*4^1 = 2^4. If the e_i are the set {1,2} we get A046951, the number of representations as a product of a number and a square. - R. J. Mathar, Nov 05 2016

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 274-278.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.12, p. 468.

E. Molnár, Sui mosaici dello spazio di dimensione n, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 51 (1971), 177-185.

J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. Speiser, Die Theorie der Gruppen von endlicher Ordnung, 4. Auflage, Birkhaeuser, 1956.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

I. G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34.

I. G. Connell, Letter to N. J. A. Sloane, no date

P. Erdős and G. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und ueber ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.

S. R. Finch, Abelian Group Enumeration Constants [broken link?]

P. Horak, Error-correcting codes and Minkowski's conjecture, Tatra Mt. Math. Publ., 45 (2010), p. 40.

B. Horvat, G. Jaklic and T. Pisanski, On the number of Hamiltonian groups, arXiv:math/0503183 [math.CO], 2005.

D. G. Kendall, R. A. Rankin, On the number of Abelian groups of a given order, Q. J. Math. 18 (1947) 197-208.

Kurokawa, Nobushige; Wakayama, Masato. Zeta extensions. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 7, 126--130. MR1930216 (2003h:11112).

H.-E. Richert, Über die Anzahl Abelscher Gruppen gegebener Ordnung I, Math. Zeitschr. 56 (1952) 21-32.

Marko Riedel, Counting Abelian Groups, Math StackExchange, October 2014.

Laszlo Toth, A note on the number of abelian groups of a given order, arXiv:1203.6473 [math.NT], (2012).

Eric Weisstein's World of Mathematics, Abelian Group

Eric Weisstein's World of Mathematics, Finite Group

Eric Weisstein's World of Mathematics, Kronecker Decomposition Theorem

Index entries for sequences related to groups

Index entries for "core" sequences

FORMULA

Multiplicative with a(p^k) = number of partitions of k = A000041(k); a(mn) = a(m)a(n) if (m, n) = 1.

a(n) = product(A000041(e(j)),j = 1..N(n)), n >= 2, if

  n  = product(prime(j)^e(j), j = 1..N(n)), N(n) = A001221(n). See the Richert reference, quoting A. Speiser's book on finite groups (in German, p. 51 in words). - Wolfdieter Lang, Jul 23 2011

In terms of the cycle index of the symmetric group: prod_{q=1}^m [z^{v_q}] Z(S_v) 1/(1-z) where v is the maximum exponent of any prime in the prime factorization of n, v_q are the exponents of the prime factors, and Z(S_v) is the cycle index of the symmetric group on v elements. - Marko Riedel, Oct 03 2014

Dirichlet g.f.: sum_{n >= 1} a(n)/n^s = product_{k >= 1} zeta(ks) [Kendall]. - Álvar Ibeas, Nov 05 2014

a(n)=2 for all n in A054753 and for all n in A085987.  a(n)=3 for all n in A030078 and for all n in A065036.  a(n)=4 for all n in A085986.  a(n)=5 for all n in A030514 and for all n in A178739.  a(n)=6 for all n in A143610. - R. J. Mathar, Nov 05 2016

EXAMPLE

a(1) = 1 since the trivial group {e} is the only group of order 1, and it is Abelian; alternatively, since the only factorization of 1 into prime powers is the empty product.

a(p) = 1 for any prime p, since the only factorization into prime powers is p = p^1, and (in view of Lagrange's theorem) there is only one group of prime order p; it is isomorphic to (Z/pZ,+) and thus Abelian.

From Wolfdieter Lang, Jul 22 2011: (Start)

a(8) = 3 because 8 = 2^3, hence a(8) = pa(3) = A000041(3) = 3 from the partitions (3), (2, 1) and (1, 1, 1), leading to the 3 factorizations of 8: 8, 4*2 and 2*2*2.

a(36) = 4 because 36 = 2^2*3^2, hence a(36) = pa(2)*pa(2) = 4 from the partitions (2) and (1, 1), leading to the 4 factorizations of 36: 2^2*3^2, 2^2*3^1*3^1, 2^1*2^1*3^2 and 2^1*2^1*3^1*3^1.

(End)

MAPLE

with(combinat): readlib(ifactors): for n from 1 to 120 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d, `, ans): od: # James A. Sellers, Dec 07 2000

MATHEMATICA

f[n_] := Times @@ PartitionsP /@ Last /@ FactorInteger@n; Array[f, 107] (* Robert G. Wilson v, Sep 22 2006 *)

Table[FiniteAbelianGroupCount[n], {n, 200}] (* Requires version 7.0 or later. - Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)

PROG

(PARI) A000688(n) = {local(f); f=factor(n); prod(i=1, matsize(f)[1], numbpart(f[i, 2]))} \\ Michael B. Porter, Feb 08 2010

(PARI) a(n)=my(f=factor(n)[, 2]); prod(i=1, #f, numbpart(f[i])) \\ Charles R Greathouse IV, Apr 16 2015

(Sage)

def a(n):

    F=factor(n)

    return prod([number_of_partitions(F[i][1]) for i in range(len(F))])

# Ralf Stephan, Jun 21 2014

(Haskell)

a000688 = product . map a000041 . a124010_row

-- Reinhard Zumkeller, Aug 28 2014

CROSSREFS

Cf. A000001, A000041, A000961, A001055, A034382, A046054, A046055, A046056, A050360, A055653, A124010, A050361, A051532, A129667 (Dirichlet inverse).

Sequence in context: A008479 A227350 A107345 * A038538 A088529 A267116

Adjacent sequences:  A000685 A000686 A000687 * A000689 A000690 A000691

KEYWORD

nonn,core,easy,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 29 19:06 EDT 2017. Contains 285613 sequences.