A061034 Maximal number of subgroups in an Abelian group with n elements.

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

Offset: 1

Author

Ola Veshta (olaveshta(AT)my-deja.com), May 26 2001

Keywords

• nonn
• mult

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.

Examples

			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.
		

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. 1-6. 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), 192-194.

Crossrefs

Cf. A006116, A018216, A083573.

Programs

PARI

{ A061034(n) = my(f=factorint(n)); prod(i=1,#f~, vecmax( apply( x->numsubgrp(f[i,1],x), partitions(f[i,2]) ) ) ); } \\ See Alekseyev link for numsubgrp(), Max Alekseyev, 2008

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

Extensions

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