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 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 (list; graph; refs; listen; history; text; internal format)
 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. 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. 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, k-i, 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, n-i, 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)my-deja.com), May 26 2001 EXTENSIONS More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003 STATUS approved

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