A000688 Number of Abelian groups of order n; number of factorizations of n into prime powers.
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
Offset: 1
Examples
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)
References
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 274-278.
- D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.12, p. 468.
- 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, Birkhäuser, 1956.
Links
- T. D. Noe, Table of n, a(n) for n = 1..10000
- Tak-Shing T. Chan, and Y.-H. Yang, Polar n-Complex and n-Bicomplex Singular Value Decomposition and Principal Component Pursuit, IEEE Transactions on Signal Processing ( Volume: 64, Issue: 24, Dec.15, 15 2016 ); DOI: 10.1109/TSP.2016.2612171.
- 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 über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102.
- Steven R. Finch, Abelian Group Enumeration Constants [Broken link]
- Steven R. Finch, Abelian Group Enumeration Constants [broken link?] [From the Wayback machine]
- 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.
- Nobushige Kurokawa and Masato Wakayama, Zeta extensions. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 7, 126--130. MR1930216 (2003h:11112).
- 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.
- H.-E. Richert, Über die Anzahl Abelscher Gruppen gegebener Ordnung I, Math. Zeitschr. 56 (1952) 21-32.
- Marko Riedel, Counting Abelian Groups, Mathematics Stack Exchange, 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
Crossrefs
Programs
-
Haskell
a000688 = product . map a000041 . a124010_row -- Reinhard Zumkeller, Aug 28 2014
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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 Sellers, Dec 07 2000
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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 *)
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PARI
A000688(n)=local(f);f=factor(n);prod(i=1,matsize(f)[1],numbpart(f[i,2])) \\ Michael B. Porter, Feb 08 2010
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PARI
a(n)=my(f=factor(n)[,2]); prod(i=1,#f,numbpart(f[i])) \\ Charles R Greathouse IV, Apr 16 2015
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Python
from sympy import factorint, npartitions from math import prod def A000688(n): return prod(map(npartitions,factorint(n).values())) # Chai Wah Wu, Jan 14 2022
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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
Formula
Multiplicative with a(p^k) = number of partitions of k = A000041(k); a(mn) = a(m)a(n) if (m, n) = 1.
a(2n) = A101872(n).
a(n) = Product_{j = 1..N(n)} A000041(e(j)), n >= 2, if
n = Product_{j = 1..N(n)} prime(j)^e(j), 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: Product_{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
From Amiram Eldar, Nov 01 2020: (Start)
a(n) = a(A057521(n)).
Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = A021002. (End)
a(n) = A005361(n) except when n is a term of A046101, since A000041(x) = x for x <= 3. - Miles Englezou, Feb 17 2024
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