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

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

Author

Jianing Song, Oct 19 2022

Keywords

• nonn
• hard

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, ...

Examples

			a(2) = 1 since there is a unique group of order A060702(2) = 6 with trivial center: S3.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..372
• Jianing Song, Number of centerless groups of order n <= 2022 (skipping n = 768, 1152, 1280, 1536, 1728, 1792, 1920, 2016)

Crossrefs

Cf. A060702, A059806, A056867.

Programs

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;