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A063966 Number of Abelian groups of order <= n.

%I A063966 #23 Jul 02 2025 16:02:01
%S A063966 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,
%T A063966 40,43,45,46,47,48,55,56,57,58,62,63,64,65,68,69,70,71,73,75,76,77,82,
%U A063966 84,86,87,89,90,93,94,97,98,99,100,102,103,104,106,117,118,119,120,122
%N A063966 Number of Abelian groups of order <= n.
%H A063966 Vaclav Kotesovec, <a href="/A063966/b063966.txt">Table of n, a(n) for n = 1..10000</a>
%H A063966 Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, <a href="https://hrcak.srce.hr/clanak/1339">On the number of hamiltonian groups</a>, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; <a href="https://arxiv.org/abs/math/0503183">arXiv preprint</a>, arXiv:math/0503183 [math.CO], 2005.
%H A063966 Hong-Quan Liu, <a href="https://eudml.org/doc/206551">On the number of abelian groups of a given order (supplement)</a>, Acta Arithmetica, Vol. 64, No. 3 (1993), pp. 285-296.
%H A063966 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbelianGroup.html">Abelian Group</a>.
%F A063966 a(n) ~ c * n, where c = A021002 = Product_{k>=2} zeta(k). - _Vaclav Kotesovec_, Oct 26 2019
%F A063966 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
%p A063966 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:
%t A063966 Accumulate[Table[FiniteAbelianGroupCount[n], {n, 1, 200}]] (* _Geoffrey Critzer_, Dec 28 2014 *)
%Y A063966 Partial sums of A000688.
%Y A063966 Cf. A063756.
%Y A063966 Cf. A021002, A084892, A084893.
%K A063966 nonn
%O A063966 1,2
%A A063966 Ahmed Fares (ahmedfares(AT)my-deja.com), Sep 04 2001
%E A063966 More terms from _James Sellers_, Sep 26 2001