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A060652 Orders of non-Abelian groups. 6
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 (list; graph; refs; listen; history; text; internal format)
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.

What exactly are these numbers? - N. J. A. Sloane, Oct 09 2006

The non-Abelian orders (or non-Abelian numbers): n such that some group of order n is non-Abelian.

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

LINKS

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

FORMULA

Let the prime factorization of n be p1^e1...pr^e^r. 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

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.

Cf. A027748, A124010, A212793.

Sequence in context: A176525 A065985 A233421 * A020739 A064466 A026286

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

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Last modified December 12 12:30 EST 2019. Contains 329958 sequences. (Running on oeis4.)