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Search: id:A061034
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| A061034 |
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Maximal number of subgroups in an abelian group with n elements. |
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+0
3
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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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.
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REFERENCES
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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]
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LINKS
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Max Alekseyev, PARI scripts for various problems
G. A. Miller, On the Subgroups of an Abelian Group [JSTOR]
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FORMULA
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(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). - njas, Dec 01 2007
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EXAMPLE
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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.
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PROGRAM
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(PARI, from Max Alekseyev) { A061034(n) = local(f=factorint(n)); prod(i=1, matsize(f)[1], A061034pp(f[i, 1], f[i, 2]) ) }
\\ for prime power p^k { A061034pp(p, k) = res=0; for(i=1, k, aux_part(p, k-i, i, [])); res }
\\ 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)); ); }
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CROSSREFS
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Cf. A006116, A018216, A083573.
Sequence in context: A108053 A133501 A124316 this_sequence A111861 A004543 A076200
Adjacent sequences: A061031 A061032 A061033 this_sequence A061035 A061036 A061037
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KEYWORD
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nonn,mult
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AUTHOR
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Ola Veshta (olaveshta(AT)my-deja.com), May 26 2001
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EXTENSIONS
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More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003
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