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

1,2

Links

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

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

Mathematica

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

Prog

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

Crossrefs

Bisection of A000688.

Cf. also A101876 (bisection of this sequence).

Cf. A000041, A021002, A048651.

Sequence in context: A340785 A355758 A206778 * A251659 A174892 A317656

Adjacent sequences: A101869 A101870 A101871 * A101873 A101874 A101875

Keyword

nonn,mult,easy

Author

N. J. A. Sloane, Jan 28 2005

Extensions

More terms from Joshua Zucker, May 10 2006

Status

approved