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%I #21 Oct 20 2022 07:43:18
%S 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 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 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 Number of groups of order A060702(n) with trivial center.
%C 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 Jianing Song, <a href="/A357900/b357900.txt">Table of n, a(n) for n = 1..372</a>
%H 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 a(2) = 1 since there is a unique group of order A060702(2) = 6 with trivial center: S3.
%o (GAP)
%o IsNilpotentNumber := function(n) # if n > 1 is a nilpotent number, then no group of order n has trivial center; see also A056867
%o local c, omega, i, j;
%o c := PrimePowersInt( n );
%o omega := Length(c)/2;
%o for i in [1..omega] do
%o for j in [1..c[2*i]] do
%o if GcdInt(n, c[2*i-1]^j-1) > 1 then
%o return false;
%o fi;
%o od;
%o od;
%o return true;
%o end;
%o CountTrivialCenter := function(n) # returns the number of groups of order n with trivial center
%o local count, i;
%o if n > 1 and IsNilpotentNumber(n) then
%o return 0;
%o fi;
%o count := 0;
%o for i in [1..NumberSmallGroups(n)] do
%o if(Size(Center(SmallGroup(n, i))) = 1) then
%o count:=count+1;
%o fi;
%o od;
%o return count;
%o end;
%Y Cf. A060702, A059806, A056867.
%K nonn,hard
%O 1,6
%A _Jianing Song_, Oct 19 2022
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