A056867 Nilpotent numbers: n such that every group of order n is nilpotent.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, 125, 127, 128, 131, 133, 135, 137, 139

Offset: 1

Author

N. J. A. Sloane, Sep 02 2000

Keywords

• nonn
• nice
• easy

Comments

Contains exactly the numbers n for which gcd(n,|A153038(n)|)=1 [Pazderski]. - R. J. Mathar, Apr 03 2012

A group G of order m is nilpotent iff it has a quotient group of order m/d for each divisor d of m. - Des MacHale and Bernard Schott, Jul 15 2022

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• J. Pakianathan and K. Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
• G. Pazderski, Die Ordnungen, zu denen nur Gruppen mit gegebener Eigenschaft gehören, Archiv Math. 10 (1) (1959) 331.
• Wikipedia, Nilpotent group.

Crossrefs

Complement of A056868.

Cf. A003277, A051532, A056866.

Cf. A027748, A124010, A173557.

Programs

GAP

IsNilpotentInt := function(n)
  local f, i, j; f := PrimePowersInt(n);
  for i in [1..Length(f)/2] do
    for j in [1..f[2*i]] do
      if Gcd(f[2*i-1]^j-1, n) > 1 then return false; fi;
    od;
  od;
  return true;
end;
Filtered([1..140], IsNilpotentInt); # Gheorghe Coserea, Dec 02 2017

Mathematica

A153038[1] = 1; A153038[n_] := (x = 1; Do[p = f[[1]]; e = f[[2]]; x = x*Product[1 - p^s, {s, 1, e}], {f, FactorInteger[n]}]; x); A056867 = Select[Range[140], GCD[#, Abs[A153038[#]]] == 1 &] (* Jean-François Alcover, May 15 2012, after R. J. Mathar *)

PARI

is(n)=my(f=factor(n));for(k=1,#f[,1], for(j=1,f[k,2], if(gcd(n, f[k,1]^j-1)>1, return(0)))); 1 \\ Charles R Greathouse IV, Sep 18 2012

Formula

n is in this sequence if p^k is not congruent to 1 mod q for any primes p and q dividing n such that p^e but not p^(e+1) divides n and k <= e. - Charles R Greathouse IV, Aug 27 2012

Extensions

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001