

A060652


Orders of nonAbelian 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
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OFFSET

1,1


COMMENTS

Because of the existence of a nonAbelian 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 nonAbelian orders (or nonAbelian numbers): n such that some group of order n is nonAbelian.
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 !! (n1)
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)mydeja.com), Apr 17 2001


EXTENSIONS

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


STATUS

approved



