A055561 Numbers m such that there are precisely 3 groups of order m.

75, 363, 609, 867, 1183, 1265, 1275, 1491, 1587, 1725, 1805, 2067, 2175, 2373, 2523, 3045, 3525, 3685, 3795, 3975, 4137, 4205, 4335, 4425, 4895, 5019, 5043, 5109, 5901, 5915, 6171, 6225, 6627, 6675, 6699, 7935, 8025, 8427, 8475, 8855, 9429, 9537, 10275

Offset

1,1

Comments

Let gnu(n) (= A000001(n)) denote the "group number of n" defined in A000001 or in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), then the sequence n -> gnu(a(n)) -> gnu(gnu(a(n))) consists of 1's. - Muniru A Asiru, Nov 19 2017

From Jianing Song, Dec 05 2021: (Start)

Contains all numbers of the form k = p*q^2, where p, q are odd primes such that q == -1 (mod p) (see A350245). The 3 groups are C_(p*q^2), C_q X C_(p*q) and (C_q X C_q) : C_p, where : means semidirect product. The third group, which is the only non-abelian group of order k, can be constructed as follows: in F_q the polynomial x^(p-1) + x^(p-2) + ... + x + 1 factors into quadratic polynomials. Pick one factor x^2 + a*x + b (all factors give the same group), then (C_q X C_q) : C_p has representation <x, y, t: x^q = y^q = t^p = 1, x*y = y*x, t*x*t^(-1) = y, t*y*t^(-1) = x^(-b)*y^(-a)>.

It seems that all terms are odd. (End)

Links

Gheorghe Coserea, Table of n, a(n) for n = 1..234567, terms 1..206 from Muniru A Asiru.

H.-U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library

H.-U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.

J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.

Gordon Royle, Numbers of Small Groups

Index entries for sequences related to groups

Example

For m = 75, the 3 groups of order 75 are C75, (C5 x C5) : C3, C15 x C5 and for m = 363 the 3 groups of order 363 are C363, (C11 x C11) : C3, C33 x C11 where C is the Cyclic group of the stated order. The symbols x and : mean direct and semi-direct products respectively. - Muniru A Asiru, Oct 24 2017

Prog

(PARI)
is(n) = {
  my(p = gcd(n, eulerphi(n)), f, g);
  if (isprime(p), return(n % p^2 == 0 && isprime(gcd(p+1, n))));
  if (omega(p) != 2 || !issquarefree(n), return(0));
  f = factor(n); g = factor(p);
  1 == g[2, 1] % g[1, 1] &&
  1 == sum(k=1, matsize(f)[1], f[k, 1] % g[1, 1] == 1) &&
  1 == sum(k=1, matsize(f)[1], f[k, 1] % g[2, 1] == 1);
};
seq(N) = {
  my(a = vector(N), k=0, n=1);
  while(k < N, if(is(n), a[k++]=n); n++); a;
};
seq(43) \\ Gheorghe Coserea, Dec 12 2017

Crossrefs

Cf. A000001. Cyclic numbers A003277. Numbers m such that there are precisely k groups of order m: A054395 (k=2), this sequence (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).

A350245 is a subsequence.

Sequence in context: A158765 A226741 A223078 * A350245 A193252 A223452

Adjacent sequences: A055558 A055559 A055560 * A055562 A055563 A055564

Keyword

nonn

Author

Christian G. Bower, May 25 2000; Nov 12 2003; Feb 17 2006

Status

approved