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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 (list; graph; refs; listen; history; text; internal format)
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 non-Abelian dihedral group of order 2n for each n>2.

Cube-free 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), 631-634.

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 cube-free 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] (* Jean-Fran├žois 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]==1|v[j]%f[i, 1]==1, return(0)))); 1 \\ Charles R Greathouse IV, Feb 13 2011

(Haskell)

a051532 n = a051532_list !! (n-1)

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

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Last modified December 21 12:18 EST 2014. Contains 252320 sequences.