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A018216 Maximal number of subgroups in a group with n elements. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Eric M. Schmidt, Table of n, a(n) for n = 1..511

FORMULA

a(n)=Maximum of {A061034(n), A083573(n)}. - Lekraj Beedassy, Oct 22 2004

(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(6) = 6 because there are two groups with 6 elements: C_6 with 4 subgroups and S_3 with 6 subgroups.

PROG

(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;

end; # Charles R Greathouse IV, Dec 27 2016

CROSSREFS

Cf. A061034.

Sequence in context: A240081 A294339 A185291 * A059907 A024931 A256612

Adjacent sequences:  A018213 A018214 A018215 * A018217 A018218 A018219

KEYWORD

nonn,nice

AUTHOR

Ola Veshta (olaveshta(AT)my-deja.com), May 23 2001

EXTENSIONS

More terms from Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003

More terms from Eric M. Schmidt, Sep 07 2012

STATUS

approved

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Last modified February 23 15:47 EST 2018. Contains 299581 sequences. (Running on oeis4.)