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A018216
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Maximal number of subgroups in a group with n elements.
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8
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1, 2, 2, 5, 2, 6, 2, 16, 6, 8, 2, 16, 2, 10, 4, 67, 2, 28, 2, 22, 10, 14, 2, 54, 8, 16, 28, 28, 2, 28, 2, 374, 4, 20, 4, 78, 2, 22, 16, 76, 2, 36, 2, 40, 12, 26, 2, 236, 10, 64, 4, 46, 2, 212, 14, 98, 22, 32, 2, 80, 2, 34, 36, 2825, 4, 52, 2, 58, 4, 52, 2, 272
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OFFSET
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1,2
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COMMENTS
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For n >= 2 a(n)>=2 with equality iff n is prime.
The minimal number of subgroups is A000005, the number of divisors of n, attained by the cyclic group of order n. - Charles R Greathouse IV, Dec 27 2016
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LINKS
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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). - N. J. A. Sloane, Dec 01 2007
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EXAMPLE
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a(6) = 6 because there are two groups with 6 elements: C_6 with 4 subgroups and S_3 with 6 subgroups.
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PROG
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(GAP) a:=function(n)
local gr, mx, t, g;
mx := 0;
gr := AllSmallGroups(n);
for g in gr do
t := Sum(ConjugacyClassesSubgroups(g), Size);
mx := Maximum(mx, t);
od;
return mx;
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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Ola Veshta (olaveshta(AT)my-deja.com), May 23 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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STATUS
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approved
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