A063966 Number of Abelian groups of order <= n.

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 25, 26, 28, 29, 31, 32, 33, 34, 37, 39, 40, 43, 45, 46, 47, 48, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 70, 71, 73, 75, 76, 77, 82, 84, 86, 87, 89, 90, 93, 94, 97, 98, 99, 100, 102, 103, 104, 106, 117, 118, 119, 120, 122

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, On the number of hamiltonian groups, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; arXiv preprint, arXiv:math/0503183 [math.CO], 2005.

Hong-Quan Liu, On the number of abelian groups of a given order (supplement), Acta Arithmetica, Vol. 64, No. 3 (1993), pp. 285-296.

Eric Weisstein's World of Mathematics, Abelian Group.

Formula

a(n) ~ c * n, where c = A021002 = Product_{k>=2} zeta(k). - Vaclav Kotesovec, Oct 26 2019

More accurately, a(n) = A021002 * n + A084892 * n^(1/2) + A084893 * n^(1/3) + O(n^(50/199 + eps)), where eps>0 is arbitrarily small (Liu, 1993). - Amiram Eldar, Sep 23 2023

Maple

with(combinat): readlib(ifactors): total := 0: for n from 1 to 100 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d, `, total+ans): total := total+ans: od:

Mathematica

Accumulate[Table[FiniteAbelianGroupCount[n], {n, 1, 200}]] (* Geoffrey Critzer, Dec 28 2014 *)

Crossrefs

Partial sums of A000688.

Cf. A063756.

Cf. A021002, A084892, A084893.

Sequence in context: A094180 A116545 A140661 * A123030 A284836 A267300

Adjacent sequences: A063963 A063964 A063965 * A063967 A063968 A063969

Keyword

nonn

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Sep 04 2001

Extensions

More terms from James A. Sellers, Sep 26 2001

Status

approved