

A061034


Maximal number of subgroups in an Abelian group with n elements.


3



1, 2, 2, 5, 2, 4, 2, 16, 6, 4, 2, 10, 2, 4, 4, 67, 2, 12, 2, 10, 4, 4, 2, 32, 8, 4, 28, 10, 2, 8, 2, 374, 4, 4, 4, 30, 2, 4, 4, 32, 2, 8, 2, 10, 12, 4, 2, 134, 10, 16, 4, 10, 2, 56, 4, 32, 4, 4, 2, 20, 2, 4, 12, 2825, 4, 8, 2, 10, 4, 8, 2, 96, 2, 4, 16, 10, 4, 8, 2, 134, 212, 4, 2
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OFFSET

1,2


COMMENTS

a(n) is multiplicative: if m and n are relatively prime then a(m*n) = a(n) * a(m). For n >= 2, a(n)>=2 with equality iff n is prime.


LINKS

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI scripts for various problems
G. A. Miller, On the Subgroups of an Abelian Group, The Annals of Mathematics, 2nd Ser., Vol. 6, No. 1. (1904), pp. 16. doi:10.2307/2007151 [See paragraph 4 entitled "Total number of subgroups in a group of order p^m".  M. F. Hasler, Dec 03 2007]
G. A. Miller, Determination of the number of subgroups of an abelian group, Bull. Amer. Math. Soc. 33 (1927), 192194.


FORMULA

(C_2)^m has A006116(m) subgroups, so this is a lower bound if n is a power of 2 (e.g. a(16) >= 67).  N. J. A. Sloane, Dec 01 2007


EXAMPLE

a(16) = 67: C16 has 5 subgroups, C2 X C8 has 11 subgroups, (C2)^2 X C4 has 27 subgroups, (C2)^4 has 67 subgroups, (C4)^2 has 15 subgroups.


PROG

(PARI) A061034(n) = local(f=factorint(n)); prod(i=1, matsize(f)[1], A061034pp(f[i, 1], f[i, 2]) )
A061034pp(p, k) = res=0; for(i=1, k, aux_part(p, ki, i, [])); res \\ for prime power p^k
aux_part(p, n, m, v) = v = concat(v, m); if(n, for(i=1, min(m, n), aux_part(p, ni, i, v)), res=max(res, numsubgrp(p, v)); ) \\ iterate over all partitions
\\ See Alekseyev link for numsubgrp.
\\ Max Alekseyev, c. 2008


CROSSREFS

Cf. A006116, A018216, A083573.
Sequence in context: A254176 A257090 A124316 * A245635 A111861 A004543
Adjacent sequences: A061031 A061032 A061033 * A061035 A061036 A061037


KEYWORD

nonn,mult


AUTHOR

Ola Veshta (olaveshta(AT)mydeja.com), May 26 2001


EXTENSIONS

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003


STATUS

approved



