A201733 Number of isomorphism classes of polycyclic groups (or solvable groups) of order n.

1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 12, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1

Offset: 1

Author

W. Edwin Clark, Dec 04 2011

Keywords

• nonn

Comments

For finite groups solvable is equivalent to polycyclic.

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..500
• Wikipedia, Polycyclic group
• Wikipedia, Solvable group

Programs

GAP

a:=[];;
N:=120;;
for n in [1..N] do
a[n]:=0;;
for j in [1..NrSmallGroups(n)] do
   if IsPcGroup(SmallGroup(n,j)) = true then
    a[n]:=a[n]+1;
   fi;
  od;
  Print(a[n],",");
od;

Formula

a(n) = A000001(n) for n < 60.

a(n) <= A000001(n) with equality if and only if n is not in A056866. In particular a(n) = A000001(n) for odd n (this is the Feit-Thompson theorem). - Benoit Jubin, Mar 30 2012