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%I #26 Jul 19 2022 05:54:37
%S 6,8,10,12,14,16,18,20,21,22,24,26,27,28,30,32,34,36,38,39,40,42,44,
%T 46,48,50,52,54,55,56,57,58,60,62,63,64,66,68,70,72,74,75,76,78,80,81,
%U 82,84,86,88,90,92,93,94,96,98,100,102,104,105,106,108,110,111,112,114,116
%N Orders of non-Abelian groups: n such that some group of order n is non-Abelian.
%C 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.
%H T. D. Noe, <a href="/A060652/b060652.txt">Table of n, a(n) for n = 1..1000</a>
%F 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
%F 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
%t 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]];
%t Select[Range[200], !abelianQ[#]&] (* _Jean-François Alcover_, Jul 19 2022, after _Charles R Greathouse IV_ *)
%o (Haskell)
%o a060652 n = a060652_list !! (n-1)
%o a060652_list = filter h [1..] where
%o h x = any (> 2) (map snd pfs) || any (== 1) pks where
%o pks = [p ^ k `mod` q | (p,e) <- pfs, q <- map fst pfs, k <- [1..e]]
%o pfs = zip (a027748_row x) (a124010_row x)
%o -- _Reinhard Zumkeller_, Jun 28 2013
%o (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
%Y Complement of A051532.
%Y Union of A056868 and A046099. - _Reinhard Zumkeller_, Jun 28 2013
%Y Cf. A027748, A124010, A212793.
%K nonn
%O 1,1
%A Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 17 2001
%E More terms from _T. D. Noe_, Mar 11 2007
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