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

1,6

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, ...

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)

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

Cf. A060702, A059806, A056867.

Sequence in context: A257679 A056059 A355915 * A357732 A356428 A158819

Adjacent sequences: A357897 A357898 A357899 * A357901 A357902 A357903

Keyword

nonn,hard

Author

Jianing Song, Oct 19 2022

Status

approved