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A059829
Maximal size of a minimal-generating-set of G where G is a finite group of order n.
2
0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 1, 4, 1, 3, 1, 2, 2, 2, 1, 3, 2, 2, 3, 2, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 4, 2, 3, 1, 2, 1, 4, 2, 3, 2, 2, 1, 2, 1, 2, 2, 6, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2
OFFSET
1,4
COMMENTS
a(n) <= floor(log_2(n)) with equality if n=2^m is a power of 2.
For n >= 2, a(n) = 1 iff n belongs to sequence A003277.
a(n) >= A051903(n). - Álvar Ibeas, Mar 28 2015
a(p^2) = 2 for all primes p, since there are only two groups (up to isomorphism) of order p^2: Z_p^2 and Z_p X Z_p. The latter is generated by 2 elements. - Tom Edgar, Apr 06 2015
LINKS
Jinyuan Wang, Table of n, a(n) for n = 1..511 (terms 1..150 from Robert Israel)
EXAMPLE
Up to isomorphism, there are five groups of order 8: the two non-abelian groups (the dihedral group and the quaternion group) are both generated by two elements, and the three abelian groups are Z_8 (generated by 1 element), Z_2 X Z_4 (generated by 2 elements) and Z_2 X Z_2 X Z_2 (generated by 3 elements). Thus a(8) = 3.
PROG
(GAP)
A := [0];
for n in [2 .. 100] do
G := AllSmallGroups(n);
m := NumberSmallGroups(n);
t := 1;
for i in [ 1 .. m] do
while EulerianFunction(G[i], t) = 0 do
t:= t+1;
od;
od;
A[n]:= t;
od;
A; # Robert Israel, Apr 01 2015
CROSSREFS
Sequence in context: A256106 A077480 A327524 * A363369 A304465 A304687
KEYWORD
nonn
AUTHOR
Noam Katz (noamkj(AT)hotmail.com), Feb 25 2001
EXTENSIONS
Offset and first term corrected by Álvar Ibeas, Mar 27 2015
More terms from Robert Israel, Apr 01 2015
STATUS
approved