

A051532


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


33



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, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 153, 157, 159, 161
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OFFSET

1,2


COMMENTS

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 nonAbelian dihedral group of order 2n for each n>2.
Cubefree terms of A056867; A212793(a(n)) = 1.  Reinhard Zumkeller, Jun 28 2013


REFERENCES

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


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, AugustSeptember 2000, pp. 631634.
Index entries for sequences related to groups


FORMULA

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


EXAMPLE

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


MATHEMATICA

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]];
Select[Range[200], okQ] (* JeanFrançois Alcover, May 03 2012, after PARI, updated Jan 10 2020 *)


PROG

(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
(Haskell)
a051532 n = a051532_list !! (n1)
a051532_list = filter ((== 1) . a212793) a056867_list
 Reinhard Zumkeller, Jun 28 2013


CROSSREFS

Subsequence of A056867; complement of A060652.
Cf. A003277, A064899.
Sequence in context: A092755 A032515 A024926 * A325461 A135785 A262249
Adjacent sequences: A051529 A051530 A051531 * A051533 A051534 A051535


KEYWORD

nonn,nice,easy


AUTHOR

Des MacHale, Dec 11 1999


STATUS

approved



