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A357900
Number of groups of order A060702(n) with trivial center.
3
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, 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, 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
OFFSET
1,6
COMMENTS
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, ...
EXAMPLE
a(2) = 1 since there is a unique group of order A060702(2) = 6 with trivial center: S3.
PROG
(GAP)
IsNilpotentNumber := function(n) # if n > 1 is a nilpotent number, then no group of order n has trivial center; see also A056867
local c, omega, i, j;
c := PrimePowersInt( n );
omega := Length(c)/2;
for i in [1..omega] do
for j in [1..c[2*i]] do
if GcdInt(n, c[2*i-1]^j-1) > 1 then
return false;
fi;
od;
od;
return true;
end;
CountTrivialCenter := function(n) # returns the number of groups of order n with trivial center
local count, i;
if n > 1 and IsNilpotentNumber(n) then
return 0;
fi;
count := 0;
for i in [1..NumberSmallGroups(n)] do
if(Size(Center(SmallGroup(n, i))) = 1) then
count:=count+1;
fi;
od;
return count;
end;
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Jianing Song, Oct 19 2022
STATUS
approved