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A181941 Numbers n such that some group of order n has a non-cyclic commutator group. 0
12, 18, 24, 32, 36, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 98, 100, 108, 112, 120, 126, 128, 132, 144, 147, 150, 156, 160, 162, 168, 180, 192, 196, 198, 200, 204, 216, 224, 225, 228, 234, 240, 242, 243, 250, 252, 256, 264, 270, 276, 280, 288, 294, 300, 306, 312, 320, 324, 336, 338, 342 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The complementary sequence 1, 2, 3, 4... is much denser and contains all n such that each group of order n has a cyclic commutator group.

Let the factorization of n into powers of squarefree mutually coprime numbers n_1, n_2, n_3, n_4, n_5,... be n = n_1 *n_2^2 *n_3^3 *n_4^4 * n_5^5*..., see A051903.

Then the complementary sequence contains n of the form n = n_1*n_2^2*n_3^3*n_4^4 under the constraints:

(i) n_4=1 or n_4=2

(ii) gcd(n, psi(n_2^2*n_3^3*n_4^4)) =1 where psi(k) = abs(A153038(k)) .

LINKS

Table of n, a(n) for n=1..64.

G. Pazderski, Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehören, Archiv Math. 10 (1) (1959) 331.

EXAMPLE

1) Does not contain 10 = 10*1*1*1 where n_4=1 and gcd(10,|A153038(1)|)=1.

Both groups of order 10 have cyclic commutator groups: D10 has C5 and C10 has E.

2) Contains 12 = 3*2^2 where n_4=1 and gcd(12,|A153038(4)|) >1.

The group A4 of order 12 has a commutator group C2 x C2 which is not cyclic.

3) Contains 18 = 2*3^2 where n_4=1 and gcd(18,|A153038(9)|) >1.

The group (C3 x C3) : C2 of order 18 has a commutator group C3 x C3 which is not cyclic. (Gap notation, SmallGroup(18,4), where the colon is the semidirect product)

4) Contains 24 = 3*1*2^3 where n_4=1 and gcd(24,|A153038(9)|) >1.

5) Contains 32 = 1*1*1*1*2^5 where n_5>1.

6) Contains 48 = 3*1*1*2^4 where n_4=2 and gcd(48,|A153038(16)|)>1.

MAPLE

nsq := proc(n)

    local f, L ;

    L := [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] ;

    if n = 1 then

        return L;

    else

        for f in ifactors(n)[2] do

            p := op(1, f) ;

            e := op(2, f) ;

            i := e ;

            L := subsop(i=op(i, L)*p^e, L) ;

        end do:

        return L ;

    end if;

end proc:

Pazdn4 := proc(L)

    if nops(L) <4 then

        1;

    else

        sqrt(sqrt(op(4, L))) ;

    end if;

end proc:

hihno1 := proc(L)

    i := 0 ;

    for j from 1 to nops(L) do

        if op(j, L) > 1 then

            i := j ;

        end if;

    end do:

    i ;

end proc:

for n from 1 to 600 do

    nf := nsq(n) ;

    n4 := Pazdn4(nf) ;

    psarg := op(2, nf)*op(3, nf)*op(4, nf) ;

    if  ( n4 =1 or n4 =2)  and gcd(n, abs(A153038(psarg))) = 1 and hihno1(nf) < 5 then

        ;

    else

        printf("%d, ", n) ;

    end if;

end do:

CROSSREFS

Sequence in context: A159886 A258914 A241646 * A206449 A241852 A113756

Adjacent sequences:  A181938 A181939 A181940 * A181942 A181943 A181944

KEYWORD

nonn

AUTHOR

R. J. Mathar, Apr 03 2012

STATUS

approved

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Last modified May 7 09:41 EDT 2021. Contains 343649 sequences. (Running on oeis4.)