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 A051532 The Abelian orders (or Abelian numbers): n such that every group of order n is Abelian. 20

%I

%S 1,2,3,4,5,7,9,11,13,15,17,19,23,25,29,31,33,35,37,41,43,45,47,49,51,

%T 53,59,61,65,67,69,71,73,77,79,83,85,87,89,91,95,97,99,101,103,107,

%U 109,113,115,119,121,123,127,131,133,137,139,141,143,145,149,151,153,157,159,161

%N The Abelian orders (or Abelian numbers): n such that every group of order n is Abelian.

%C Except for a(2)=2 and a(4)=4, all of the terms in the sequence are odd. This is because of the existence of a non-Abelian dihedral group of order 2n for each n>2.

%C Cubefree terms of A056867; A212793(a(n)) = 1. - _Reinhard Zumkeller_, Jun 28 2013

%D W. R. Scott, Group Theory, Dover, 1987, page 217.

%H T. D. Noe, <a href="/A051532/b051532.txt">Table of n, a(n) for n = 1..1000</a>

%H J. Pakianathan and K. Shankar, <a href="http://www.jstor.org/stable/2589118">Nilpotent Numbers</a>, Amer. Math. Monthly, 107, August-September 2000, pp. 631-634.

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F n must be cubefree and its prime divisors must satisfy certain congruences.

%F Let the prime factorization of n be p1^e1...pr^er. Then n is in this sequence if ei<3 for all i and pi^k does not equal 1 (mod pj) for all i and j and 1 <= k <= ei. - _T. D. Noe_, Mar 25 2007

%e a(4)=4 because every group of order 4 is Abelian.

%t isA051532[n_] := Catch[f = FactorInteger[n]; v = f[[All, 1]]; lv = Length[v]; For[i = 1, i <= lv, i++, If[f[[i, 2]] > 2, Throw[False], v[[i]] = f[[i, 1]]^f[[i, 2]]]]; For[i = 1, i <= lv, i++, For[j = i + 1, j <= lv, j++, If[Mod[v[[i]], f[[j, 1]]] == 1 || Mod[v[[j]], f[[i, 1]]] == 1, Throw[False]]]]; Throw[True]]; Select[Range, isA051532] (* _Jean-François Alcover_, May 03 2012, after PARI *)

%o (PARI) is(n)=my(f=factor(n),v=vector(#f[,1])); for(i=1,#v, if(f[i,2]>2, return(0), 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(0)))); 1 \\ _Charles R Greathouse IV_, Feb 13 2011

%o a051532 n = a051532_list !! (n-1)

%o a051532_list = filter ((== 1) . a212793) a056867_list

%o -- _Reinhard Zumkeller_, Jun 28 2013

%Y Subsequence of A056867; complement of A060652.

%Y Cf. A003277, A064899.

%K nonn,nice,easy

%O 1,2

%A _Des MacHale_

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Last modified March 20 07:27 EDT 2019. Contains 321345 sequences. (Running on oeis4.)