%I #8 Apr 04 2012 10:33:54
%S 12,18,24,32,36,48,50,54,56,60,64,72,75,80,81,84,90,96,98,100,108,112,
%T 120,126,128,132,144,147,150,156,160,162,168,180,192,196,198,200,204,
%U 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
%N Numbers n such that some group of order n has a non-cyclic commutator group.
%C 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.
%C 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.
%C Then the complementary sequence contains n of the form n = n_1*n_2^2*n_3^3*n_4^4 under the constraints:
%C (i) n_4=1 or n_4=2
%C (ii) gcd(n, psi(n_2^2*n_3^3*n_4^4)) =1 where psi(k) = abs(A153038(k)) .
%H G. Pazderski, <a href="http://dx.doi.org/10.1007/BF01240807">Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehören</a>, Archiv Math. 10 (1) (1959) 331.
%e 1) Does not contain 10 = 10*1*1*1 where n_4=1 and gcd(10,|A153038(1)|)=1.
%e Both groups of order 10 have cyclic commutator groups: D10 has C5 and C10 has E.
%e 2) Contains 12 = 3*2^2 where n_4=1 and gcd(12,|A153038(4)|) >1.
%e The group A4 of order 12 has a commutator group C2 x C2 which is not cyclic.
%e 3) Contains 18 = 2*3^2 where n_4=1 and gcd(18,|A153038(9)|) >1.
%e 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)
%e 4) Contains 24 = 3*1*2^3 where n_4=1 and gcd(24,|A153038(9)|) >1.
%e 5) Contains 32 = 1*1*1*1*2^5 where n_5>1.
%e 6) Contains 48 = 3*1*1*2^4 where n_4=2 and gcd(48,|A153038(16)|)>1.
%p nsq := proc(n)
%p local f,L ;
%p 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] ;
%p if n = 1 then
%p return L;
%p else
%p for f in ifactors(n)[2] do
%p p := op(1,f) ;
%p e := op(2,f) ;
%p i := e ;
%p L := subsop(i=op(i,L)*p^e,L) ;
%p end do:
%p return L ;
%p end if;
%p end proc:
%p Pazdn4 := proc(L)
%p if nops(L) <4 then
%p 1;
%p else
%p sqrt(sqrt(op(4,L))) ;
%p end if;
%p end proc:
%p hihno1 := proc(L)
%p i := 0 ;
%p for j from 1 to nops(L) do
%p if op(j,L) > 1 then
%p i := j ;
%p end if;
%p end do:
%p i ;
%p end proc:
%p for n from 1 to 600 do
%p nf := nsq(n) ;
%p n4 := Pazdn4(nf) ;
%p psarg := op(2,nf)*op(3,nf)*op(4,nf) ;
%p if ( n4 =1 or n4 =2) and gcd(n, abs(A153038(psarg))) = 1 and hihno1(nf) < 5 then
%p ;
%p else
%p printf("%d,",n) ;
%p end if;
%p end do:
%K nonn
%O 1,1
%A _R. J. Mathar_, Apr 03 2012
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