This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350422 #30 Jan 13 2022 08:19:57
%S A350422 45,99,153,175,207,245,261,325,369,423,425,475,477,531,539,575,637,
%T A350422 639,725,747,801,833,845,847,909,925,931,963,1017,1075,1127,1175,1179,
%U A350422 1233,1325,1341,1445,1475,1503,1519,1557,1573,1611,1675,1719,1773,1813,1825,1859,1975,2009
%N A350422 Numbers of the form m = p^2*q for which there exist exactly 2 groups of order m.
%C A350422 Terms come from the union of terms of the form p^2*q with p < q in A350332 and terms of the same form with p > q in A350421, with p, q odd primes.
%C A350422 All terms are odd.
%C A350422 These 2 groups are abelian; they are C_{p^2*q} and (C_p X C_p) X C_q, where C means cyclic groups of the stated order and the symbol X means direct product.
%D A350422 Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.
%e A350422 With p < q: 175 = 5^2 * 7, 5 and 7 are odd primes and 5 does not divide 7-1 = 6, hence 175 is a term (see A350332).
%e A350422 With p > q: 245 = 7^2 * 5, 5 and 7 are odd primes, 5 does not divide 7-1 = 6 and does not divide 7+1 = 8, hence 245 is a term (see A350421).
%t A350422 q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (e == {1, 2} && ! Or @@ Divisible[p[[2]] + {-1, 1}, p[[1]]]) || (e == {2, 1} && ! Divisible[p[[2]] - 1, p[[1]]])]; Select[Range[1, 2000, 2], q] (* _Amiram Eldar_, Jan 03 2022 *)
%o A350422 (PARI) isoka(f) = if (f[, 2] == [2, 1]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p)); \\ A350332
%o A350422 isokb(f) = if (f[, 2] == [1, 2]~, my(p=f[2, 1], q=f[1, 1]); ((p-1) % q) && ((p+1) % q)); \\ A350421
%o A350422 isok(m) = my(f=factor(m)); isoka(f) || isokb(f); \\ _Michel Marcus_, Jan 09 2022
%Y A350422 Disjoint union of A350332 (p<q) and A350421 (p>q).
%Y A350422 Intersection of A054395 and A054753.
%Y A350422 Subsequence of A051532, A060687 and A350322.
%Y A350422 Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245, A350638.
%K A350422 nonn
%O A350422 1,1
%A A350422 _Bernard Schott_, Jan 03 2022