

A059829


Maximal size of a minimalgeneratingset of G where G is a finite group of order n.


1



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
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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

Robert Israel, Table of n, a(n) for n = 1..150


EXAMPLE

Up to isomorphism, there are five groups of order 8: the two nonabelian 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

Cf. A003277, A051903.
Sequence in context: A256106 A077480 A327524 * A304465 A304687 A076558
Adjacent sequences: A059826 A059827 A059828 * A059830 A059831 A059832


KEYWORD

nonn,more


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



