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%I #22 Sep 23 2023 12:11:47
%S 1,2,1,3,1,2,1,5,2,2,1,3,1,2,1,7,1,4,1,3,1,2,1,5,2,2,3,3,1,2,1,11,1,2,
%T 1,6,1,2,1,5,1,2,1,3,2,2,1,7,2,4,1,3,1,6,1,5,1,2,1,3,1,2,2,15,1,2,1,3,
%U 1,2,1,10,1,2,2,3,1,2,1,7,5,2,1,3,1,2,1,5,1,4,1,3,1,2,1,11,1,4,2,6,1,2,1,5
%N Number of Abelian groups of order 2n.
%H Antti Karttunen, <a href="/A101872/b101872.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = A000688(2n).
%F Multiplicative with a(2^k) = A000041(1+k), and for odd primes p, a(p^k) = A000041(k), where A000041(k) is the number of partitions of k. - _Antti Karttunen_, Sep 27 2018
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * (1-A048651) * A021002 = 3.26425865613408900779... . - _Amiram Eldar_, Sep 23 2023
%t Table[FiniteAbelianGroupCount[2 k], {k, 1, 100}] (* _Geoffrey Critzer_, Dec 29 2014 *)
%o (PARI) A101872(n) = factorback(apply(e -> numbpart(e),factor(2*n)[,2])); \\ _Antti Karttunen_, Sep 27 2018
%Y Bisection of A000688.
%Y Cf. also A101876 (bisection of this sequence).
%Y Cf. A000041, A021002, A048651.
%K nonn,mult,easy
%O 1,2
%A _N. J. A. Sloane_, Jan 28 2005
%E More terms from _Joshua Zucker_, May 10 2006
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