A101872 Number of Abelian groups of order 2n.

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, 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, 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

Offset: 1

Author

N. J. A. Sloane, Jan 28 2005

Keywords

• nonn
• mult
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Bisection of A000688.

Cf. also A101876 (bisection of this sequence).

Cf. A000041, A021002, A048651.

Programs

Mathematica

Table[FiniteAbelianGroupCount[2 k], {k, 1, 100}] (* Geoffrey Critzer, Dec 29 2014 *)

PARI

A101872(n) = factorback(apply(e -> numbpart(e),factor(2*n)[,2])); \\ Antti Karttunen, Sep 27 2018

Formula

a(n) = A000688(2n).

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

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * (1-A048651) * A021002 = 3.26425865613408900779... . - Amiram Eldar, Sep 23 2023

Extensions

More terms from Joshua Zucker, May 10 2006