

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
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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)mydeja.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



