A054395 Numbers m such that there are precisely 2 groups of order m.

4, 6, 9, 10, 14, 21, 22, 25, 26, 34, 38, 39, 45, 46, 49, 55, 57, 58, 62, 74, 82, 86, 93, 94, 99, 105, 106, 111, 118, 121, 122, 129, 134, 142, 146, 153, 155, 158, 165, 166, 169, 175, 178, 183, 194, 195, 201, 202, 203, 205, 206, 207, 214, 218, 219, 226, 231, 237

Offset: 1

Author

N. J. A. Sloane, May 21 2000

Keywords

• nonn

Comments

Givens characterizes this sequence, see Theorem 5. In particular, this sequence is ({n: A215935(n) = 1} INTERSECT A005117) UNION (A060687 INTERSECT A051532). - Charles R Greathouse IV, Aug 27 2012 [This is now A350586 UNION A350322. - Charles R Greathouse IV, Jan 08 2022]

Numbers m such that A000001(m) = 2. - Muniru A Asiru, Nov 03 2017

Examples

			For m = 4, the 2 groups of order 4 are C4, C2 x C2; for m = 6, the 2 groups of order 6 are S3, C6; and for m = 9, the 2 groups of order 9 are C9, C3 x C3 where C is the cyclic group of the stated order and S is the symmetric group of the stated degree. The symbol x means direct product. - _Muniru A Asiru_, Oct 24 2017
		

Links

• Muniru A Asiru and Gheorghe Coserea, Table of n, a(n) for n = 1..234567, terms 1..422 from Muniru A Asiru.
• H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library
• Clint Givens, Orders for which there exist exactly two groups (2006)
• Gordon Royle, Numbers of Small Groups
• Index entries for sequences related to groups

Crossrefs

Equals A350586 UNION A350322.

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: this sequence (k=2), A055561 (k=3), A054396 (k=4), A054397 (k=5), A135850 (k=6), A249550 (k=7), A249551 (k=8), A249552 (k=9), A249553 (k=10), A249554 (k=11), A249555 (k=12), A292896 (k=13), A294155 (k=14), A294156 (k=15), A295161 (k=16), A294949 (k=17), A298909 (k=18), A298910 (k=19), A298911 (k=20).

Programs

GAP

A054395 := Filtered([1..2015], n -> NumberSmallGroups(n) = 2); # Muniru A Asiru, Oct 24 2017

GAP

IsGivensInt := function(n)
  local p, f; p := GcdInt(n, Phi(n));
  if not IsPrimeInt(p) then return false; fi;
  if n mod p^2 = 0 then return 1 = GcdInt(p+1, n); fi;
  f := PrimePowersInt(n);
  return 1 = Number([1..QuoInt(Length(f),2)], k->f[2*k-1] mod p = 1);
end;;
Filtered([1..240], IsGivensInt); # Gheorghe Coserea, Dec 04 2017

Mathematica

Select[Range[240], FiniteGroupCount[#] == 2&]
(* or: *)
okQ[n_] := Module[{p, f}, p = GCD[n, EulerPhi[n]]; If[! PrimeQ[p], Return[False]]; If[Mod[n, p^2] == 0, Return[1 == GCD[p + 1, n]]]; f = FactorInteger[n]; 1 == Sum[Boole[Mod[f[[k, 1]], p] == 1], {k, 1, Length[f]}]];
Select[Range[240], okQ] (* Jean-François Alcover, Dec 08 2017, after Gheorghe Coserea *)

PARI

is(n) = {
  my(p=gcd(n,eulerphi(n)), f);
  if (!isprime(p), return(0));
  if (n%p^2 == 0, return(1 == gcd(p+1, n)));
  f = factor(n); 1 == sum(k=1, matsize(f)[1], f[k,1]%p==1);
};
seq(N) = {
  my(a = vector(N), k=0, n=1);
  while(k < N, if(is(n), a[k++]=n); n++); a;
};
seq(58) \\ Gheorghe Coserea, Dec 03 2017

Extensions

More terms from Christian G. Bower, May 25 2000