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A357900 Number of groups of order A060702(n) with trivial center.

%I A357900 #21 Oct 20 2022 07:43:18
%S A357900 1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,2,2,1,5,1,1,1,1,3,1,1,1,1,6,1,1,
%T A357900 2,1,1,2,1,2,1,1,5,2,5,1,1,5,2,1,2,1,1,4,1,4,1,1,1,1,1,1,2,1,3,1,3,1,
%U A357900 4,1,1,4,1,1,17,1,1,5,1,1,1,1,8,1,1,2,1,11,1,2,2,5,1,1,1,2,1,1,3,1,1,19
%N A357900 Number of groups of order A060702(n) with trivial center.
%C A357900 Among the data currently known, it seems that the indices of records are n's such that A060702(n) = 1, 18, 54, 72, 162, 216, 486, 648, 972, 1458, ... with record values 1, 2, 5, 6, 17, 19, 72, 79, 109, 443, ...
%H A357900 Jianing Song, <a href="/A357900/b357900.txt">Table of n, a(n) for n = 1..372</a>
%H A357900 Jianing Song, <a href="/A357900/a357900.txt">Number of centerless groups of order n <= 2022 (skipping n = 768, 1152, 1280, 1536, 1728, 1792, 1920, 2016)</a>
%e A357900 a(2) = 1 since there is a unique group of order A060702(2) = 6 with trivial center: S3.
%o A357900 (GAP)
%o A357900 IsNilpotentNumber := function(n) # if n > 1 is a nilpotent number, then no group of order n has trivial center; see also A056867
%o A357900     local c, omega, i, j;
%o A357900     c := PrimePowersInt( n );
%o A357900     omega := Length(c)/2;
%o A357900     for i in [1..omega] do
%o A357900         for j in [1..c[2*i]] do
%o A357900             if GcdInt(n, c[2*i-1]^j-1) > 1 then
%o A357900                 return false;
%o A357900             fi;
%o A357900         od;
%o A357900     od;
%o A357900     return true;
%o A357900 end;
%o A357900 CountTrivialCenter := function(n) # returns the number of groups of order n with trivial center
%o A357900     local count, i;
%o A357900     if n > 1 and IsNilpotentNumber(n) then
%o A357900         return 0;
%o A357900     fi;
%o A357900     count := 0;
%o A357900     for i in [1..NumberSmallGroups(n)] do
%o A357900         if(Size(Center(SmallGroup(n, i))) = 1) then
%o A357900             count:=count+1;
%o A357900         fi;
%o A357900     od;
%o A357900     return count;
%o A357900 end;
%Y A357900 Cf. A060702, A059806, A056867.
%K A357900 nonn,hard
%O A357900 1,6
%A A357900 _Jianing Song_, Oct 19 2022