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%I #6 Sep 08 2022 08:45:29
%S 1,2,2,5,2,7,2,17,5,7,2,23,2,7,4,67,2,23,2,25,8,7,2,99,5,7,18,20,2,25,
%T 2,342,4,7,4,89,2,7,8,99,2,40,2,20,10,7,2,476,5,23,4,25,2,100,10,87,8,
%U 7,2,115,2,7,24,2602,4,25,2,25,4,25,2,461,2,7,13,20,4,40,2,504,79,7,2,141,4,7,4,83,2,83,4,20,8,7,4
%N Number of pointed groups of order n: that is, Sum_{G = group of order n} Number of orbits in G under the full automorphism group of G.
%C Number of pairs (G, g in G) for G a group of order n, g an orbit representative for action of Aut(G) on G.
%C This has the same relation to A000001 (groups) as A000081 (pointed trees, also called rooted trees) does to trees (A000055).
%H Klaus Brockhaus, <a href="/A126103/b126103.txt">Table of n, a(n) for n=1..191</a>
%o (Magma) D:=SmallGroupDatabase();
%o for o in [1..95] do
%o t1:=0;
%o t2:=NumberOfSmallGroups(D,o);
%o for n in [1..t2] do
%o G:=SmallGroup(D,o,n);
%o H:=AutomorphismGroup(G);
%o gg:=[];
%o for g in G do Append(~gg,g);
%o end for;
%o PH:=[];
%o for h in Generators(H) do
%o ph:=[];
%o for i in [1..#gg] do
%o j:=Position(gg,gg[i]@h);
%o Append(~ph,j);
%o end for;
%o Append(~PH,ph);
%o end for;
%o pH:=sub<SymmetricGroup(#gg) | PH>;
%o t1:=t1 + #Orbits(pH);
%o end for;
%o print(t1);
%o end for;
%o (Magma) D:=SmallGroupDatabase(); [ &+[ #Orbits(sub<SymmetricGroup(o) | [ [ Position(gg, h(gg[i])): i in [1..o] ] where gg is [g: g in G] : h in Generators(AutomorphismGroup(G)) ] where G is SmallGroup(D, o, n) > ) : n in [1..NumberOfSmallGroups(D, o)] ] : o in [1..95] ]; /* Klaus Brockhaus, Mar 08 2007 */
%Y Cf. A000001 (groups). See A126102 for a different and somewhat inferior version.
%K nonn
%O 1,2
%A Gabriele Nebe and _N. J. A. Sloane_, Mar 06 2007
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