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%I #29 May 25 2018 14:31:30
%S 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,
%T 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,
%U 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
%N Maximal number of subgroups in an Abelian group with n elements.
%C 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.
%H Eric M. Schmidt, <a href="/A061034/b061034.txt">Table of n, a(n) for n = 1..10000</a>
%H Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI scripts for various problems</a>
%H 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:<a href="http://dx.doi.org/10.2307/2007151">10.2307/2007151</a> [See paragraph 4 entitled "Total number of subgroups in a group of order p^m". - M. F. Hasler, Dec 03 2007]
%H G. A. Miller, <a href="http://dx.doi.org/10.1090/S0002-9904-1927-04354-3">Determination of the number of subgroups of an abelian group</a>, Bull. Amer. Math. Soc. 33 (1927), 192-194.
%F (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
%e 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.
%o (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
%Y Cf. A006116, A018216, A083573.
%K nonn,mult
%O 1,2
%A Ola Veshta (olaveshta(AT)my-deja.com), May 26 2001
%E More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003
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