A060652 Orders of non-Abelian groups: n such that some group of order n is non-Abelian.

6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116

Offset

1,1

Comments

Because of the existence of a non-Abelian dihedral group of order 2n for each n>2 all the even numbers >= 6 are in this sequence.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Formula

Let the prime factorization of n be p1^e1...pr^er. Then n is in this sequence if ei>2 for some i or pi^k = 1 (mod pj) for some i and j and 1 <= k <= ei. - T. D. Noe, Mar 25 2007

Equivalently: Let the prime factorization of n be p1^e1...pr^er. Then n is in this sequence if ei>2 for some i or if pi^ei = 1 (mod pj) for some i and j. - Charles R Greathouse IV, Jan 09 2022

Mathematica

abelianQ[n_] := Module[{f, lf, p, e, v}, f = FactorInteger[n]; lf = Length[f]; p = f[[All, 1]]; e = f[[All, 2]]; If[AnyTrue[e, # > 2&], Return[False]]; v = p^e; For[i = 1, i <= lf, i++, For[j = i+1, j <= lf, j++, If[Mod[v[[i]], p[[j]]] == 1 || Mod[v[[j]], p[[i]]] == 1, Return[False]]]]; Return[True]];
Select[Range[200], !abelianQ[#]&] (* Jean-François Alcover, Jul 19 2022, after Charles R Greathouse IV *)

Prog

(Haskell)
a060652 n = a060652_list !! (n-1)
a060652_list = filter h [1..] where
   h x = any (> 2) (map snd pfs) || any (== 1) pks where
     pks = [p ^ k `mod` q | (p, e) <- pfs, q <- map fst pfs, k <- [1..e]]
     pfs = zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Jun 28 2013
(PARI) is(n)=my(f=factor(n), v=vector(#f[, 1])); for(i=1, #v, if(f[i, 2]>2, return(1), v[i]=f[i, 1]^f[i, 2])); for(i=1, #v, for(j=i+1, #v, if(v[i]%f[j, 1]==1 || v[j]%f[i, 1]==1, return(1)))); 0 \\ Charles R Greathouse IV, Apr 16 2015

Crossrefs

Complement of A051532.

Union of A056868 and A046099. - Reinhard Zumkeller, Jun 28 2013

Cf. A027748, A124010, A212793.

Sequence in context: A176525 A065985 A233421 * A020739 A064466 A356609

Adjacent sequences: A060649 A060650 A060651 * A060653 A060654 A060655

Keyword

nonn

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 17 2001

Extensions

More terms from T. D. Noe, Mar 11 2007

Status

approved