%I A061034
%S A061034 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,
%T A061034 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,
%U A061034 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
%N A061034 Maximal number of subgroups in an Abelian group with n elements.
%C A061034 a(n) is multiplicative: if m and n are relatively primes then a(m*n) 
               = a(n) * a(m) . For n >= 2 a(n)>=2 with equality iff n is prime.
%D A061034 G. A. Miller, On the Subgroups of an Abelian Group, The Annals of Mathematics, 
               2nd Ser., Vol. 6, No. 1. (1904), pp. 1-6. [Paragraph 4 is entitled 
               "total number of subgroups in a group of order p^m". - M. F. Hasler, 
               Dec 03 2007]
%H A061034 Max Alekseyev, <a href="http://www.cs.ucsd.edu/users/maxal/gpscripts/
               ">PARI scripts for various problems</a>
%H A061034 G. A. Miller, <a href="http://links.jstor.org/sici?sici=0003-486X%28190410%292%3A6%3A1%3C1%3AOTSOAA%3E2.0.CO%\
               3B2-P">On the Subgroups of an Abelian Group</a> [JSTOR]
%F A061034 (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 (njas(AT)research.att.com), 
               Dec 01 2007
%e A061034 a(4) = 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 A061034 (PARI, from Max Alekseyev) { A061034(n) = local(f=factorint(n)); prod(i=1,
               matsize(f)[1], A061034pp(f[i,1],f[i,2]) ) }
%o A061034 \\ for prime power p^k { A061034pp(p,k) = res=0; for(i=1, k, aux_part(p, 
               k-i, i, [])); res }
%o A061034 \\ iterate over all partitions { aux_part(p, n, m, v) = v = concat(v,
               m); if(n, for(i=1, min(m,n), aux_part(p, n-i, i, v)), res=max(res,
               numsubgrp(p,v));); }
%Y A061034 Cf. A006116, A018216, A083573.
%Y A061034 Sequence in context: A108053 A133501 A124316 this_sequence A111861 A004543 
               A153078
%Y A061034 Adjacent sequences: A061031 A061032 A061033 this_sequence A061035 A061036 
               A061037
%K A061034 nonn,mult
%O A061034 1,2
%A A061034 Ola Veshta (olaveshta(AT)my-deja.com), May 26 2001
%E A061034 More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003