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A241276 Number of partitions of n that come from sizes of conjugacy classes of groups of order n. 1
1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 7, 1, 2, 2, 2, 1, 4, 1, 6, 1, 2, 1, 6, 1, 2, 2, 5, 1, 6, 1, 2, 1, 2, 1, 13, 1, 3, 1, 3, 1, 7, 2, 5, 2, 2, 1, 9, 1, 2, 2, 16, 1, 4, 1, 3, 1, 4, 1, 17, 1, 2, 2, 2, 1, 6, 1, 11, 3, 2, 1, 9, 1, 2, 1, 4, 1, 6, 1, 2, 2, 2, 1, 30, 1, 3, 1, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

a(n) = 1 if every group of order n is abelian, that is, if n is in A051532.

Upper bounds are given by A000001 (number of groups of order n) and A018818 (number of partitions of n into divisors of n).

A077191 is an upper bound. - Eric M. Schmidt, Oct 16 2014

LINKS

Eric M. Schmidt, Table of n, a(n) for n = 1..1023

Wikipedia, Conjugacy Class

EXAMPLE

If n = 6 there are two groups of order 6: Z_6, all of whose conjugacy classes are of order 1 giving the partition [1,1,1,1,1,1] and S_6, which has three conjugacy classes whose sizes are 1, 2 and 3, giving the partition [1,2,3]. Hence a(6) = 2.

PROG

(GAP)

a:=[];;

for n in [1..100] do

  P:=[];

  for i in [1..NumberSmallGroups(n)] do

   g:=SmallGroup(n, i);

   cc:=ConjugacyClasses(g);

   L:=List(cc, Size);

   Sort(L);

   Add(P, L);

   P:=Set(P);

  od;

  Add(a, Length(P));

od;

a;

(GAP) a := function(n) local i, p, P; P := []; for i in [1..NrSmallGroups(n)] do p := List(ConjugacyClasses(SmallGroup(n, i)), Size); Sort(p); MakeImmutable(p); AddSet(P, p); od; return Length(P); end; # Eric M. Schmidt, Oct 16 2014

CROSSREFS

Sequence in context: A119647 A087942 A320012 * A292286 A099042 A140774

Adjacent sequences:  A241273 A241274 A241275 * A241277 A241278 A241279

KEYWORD

nonn,hard

AUTHOR

W. Edwin Clark, Apr 18 2014

STATUS

approved

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Last modified February 22 05:17 EST 2019. Contains 320385 sequences. (Running on oeis4.)