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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 G. Pazderski, Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehĂ¶ren, 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.)