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%I #63 May 13 2023 23:50:09
%S 75,363,609,867,1183,1265,1275,1491,1587,1725,1805,2067,2175,2373,
%T 2523,3045,3525,3685,3795,3975,4137,4205,4335,4425,4895,5019,5043,
%U 5109,5901,5915,6171,6225,6627,6675,6699,7935,8025,8427,8475,8855,9429,9537,10275
%N Numbers m such that there are precisely 3 groups of order m.
%C 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
%C From _Jianing Song_, Dec 05 2021: (Start)
%C 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)>.
%C It seems that all terms are odd. (End)
%H Gheorghe Coserea, <a href="/A055561/b055561.txt">Table of n, a(n) for n = 1..234567</a>, terms 1..206 from Muniru A Asiru.
%H H.-U. Besche, B. Eick and E. A. O'Brien, <a href="http://www.icm.tu-bs.de/ag_algebra/software/small/">The Small Groups Library</a>
%H H.-U. Besche, B. Eick and E. A. O'Brien, <a href="http://dx.doi.org/10.1142/S0218196702001115">A Millennium Project: Constructing Small Groups</a>, Internat. J. Algebra and Computation, 12 (2002), 623-644.
%H J. H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf"> Counting groups: gnus, moas and other exotica</a>, Math. Intell., Vol. 30, No. 2, Spring 2008.
%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cubcay/">Numbers of Small Groups</a>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%e 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
%o (PARI)
%o is(n) = {
%o my(p = gcd(n, eulerphi(n)),f,g);
%o if (isprime(p), return(n % p^2 == 0 && isprime(gcd(p+1, n))));
%o if (omega(p) != 2 || !issquarefree(n), return(0));
%o f = factor(n); g = factor(p);
%o 1 == g[2,1] % g[1,1] &&
%o 1 == sum(k=1, matsize(f)[1], f[k,1] % g[1,1] == 1) &&
%o 1 == sum(k=1, matsize(f)[1], f[k,1] % g[2,1] == 1);
%o };
%o seq(N) = {
%o my(a = vector(N), k=0, n=1);
%o while(k < N, if(is(n), a[k++]=n); n++); a;
%o };
%o seq(43) \\ _Gheorghe Coserea_, Dec 12 2017
%Y 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).
%Y A350245 is a subsequence.
%K nonn
%O 1,1
%A _Christian G. Bower_, May 25 2000; Nov 12 2003; Feb 17 2006
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