

A055561


Numbers n such that there are precisely 3 groups of order n.


25



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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


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), 623644.
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 n = 75, the 3 groups of order 75 are C75, (C5 x C5) : C3, C15 x C5 and for n = 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 semidirect 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, A003277, A054395, A054396, A054397.
Sequence in context: A158765 A226741 A223078 * A193252 A223452 A015223
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



