%I #74 Oct 08 2023 04:53:39

%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): numbers m such that every group of order m 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 2m for each m > 2.

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

%C See similar comment with "squarefree terms" in A003277 (Donald J. McCarthy link). - _Bernard Schott_, Feb 20 2023

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

%H Michael S. Branicky, <a href="/A051532/b051532.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Donald J. McCarthy, <a href="https://doi.org/10.1111/j.2164-0947.1971.tb02624.x">A survey of partial converses to Lagrange's theorem on finite groups</a>, Transactions of the New York Academy of Sciences, 1971, page 592.

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

%F Let the prime factorization of m be p1^e1 * ... * pr^er. Then m is in this sequence if ei < 3 for all i and pi^k is not congruent to 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.

%e These two abelian groups of order 4 are the cyclic group C_4 and the Klein four-group = C_2 X C_2, the smallest non-cyclic abelian group. - _Bernard Schott_, Feb 21 2023

%t okQ[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], okQ] (* _Jean-François Alcover_, May 03 2012, after PARI, updated Jan 10 2020 *)

%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 (Haskell)

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

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

%o -- _Reinhard Zumkeller_, Jun 28 2013

%o (Python)

%o from sympy import factorint

%o def ok(n):

%o if n == 1: return True

%o f = factorint(n)

%o p, e = f.keys(), f.values()

%o if max(e) >= 3: return False

%o return all((pi**k)%pj!=1 for pi in p for pj in p if pj!=pi for k in range(1, f[pi]+1))

%o print([k for k in range(1, 162) if ok(k)]) # _Michael S. Branicky_, Feb 20 2023

%Y Subsequence of A056867 and supersequence of A003277.

%Y Complement of A060652.

%Y Intersection of A004709 and A056867.

%Y Cf. A064899, A212793.

%K nonn,nice,easy

%O 1,2

%A _Des MacHale_, Dec 11 1999