

A051532


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


18



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

J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. Monthly, 107 (Aug. 2000), 631634.
W. R. Scott, Group Theory, Dover, 1987, page 217.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000


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

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[200], isA051532] (* JeanFrançois Alcover, May 03 2012, after PARI *)


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]==1v[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

Cf. A003277, A064899, A060652.
Sequence in context: A092755 A032515 A024926 * A135785 A248421 A008732
Adjacent sequences: A051529 A051530 A051531 * A051533 A051534 A051535


KEYWORD

nonn,nice,easy


AUTHOR

Des MacHale


STATUS

approved



