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A349495 Numbers p^2*q, p primes such that p divides q-1 and p^2 does not divide q-1, with (p,q) <> (2,3).

%I A349495 #51 Dec 17 2021 20:38:24
%S A349495 28,44,63,76,92,117,124,172,188,236,268,275,279,284,316,332,387,412,
%T A349495 428,508,524,549,556,603,604,652,668,711,716,764,775,796,844,873,892,
%U A349495 908,927,956,1004,1025,1052,1084,1132,1228,1244,1251,1324,1359,1388,1413,1421
%N A349495 Numbers p^2*q, p<q primes such that p divides q-1 and p^2 does not divide q-1, with (p,q) <> (2,3).
%C A349495 For these terms m, there are precisely 4 groups of order m, so this is a subsequence of A054396.
%C A349495 Two of them are abelian: C_{p^2*q}, C_q X C_p X C_p, and the two others that are nonabelian are C_q : (C_p x C_p), and C_q : C_p^2. Note that when p = 2, C_q : (C_p x C_p) ~ D_{p^2*q}. Here C and D mean cyclic and dihedral groups of the stated order, the symbols ~,  X and : mean "isomorphic to", direct and semidirect products respectively.
%C A349495 Why (p,q) <> (2,3)?  Because there are 5 groups of order 12, and in this particular case, the 5th group is the alternating group A_4 because 2^2*3 = 4!/2 (see Example section in A054397).
%C A349495 Contains 4*r for r in A002145 and r > 3. - _Alois P. Heinz_, Dec 15 2021
%D A349495 Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.
%e A349495 28 = 2^2*7, 2 divides 7-1 = 6 and 2^2 does not divide 7-1 = 6, hence 28 is a term.
%e A349495 63 = 3^2*7, 3 divides 7-1 = 6 and 3^2 does not divide 7-1 = 6, hence 63 is another term.
%t A349495 q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {2, 1} && IntegerExponent[p[[2]] - 1, p[[1]]] == 1]; Select[Range[28, 1500], q]  (* _Amiram Eldar_, Dec 16 2021 *)
%o A349495 (PARI) isok(m) = if (m==12, return(0)); my(f=factor(m)); if (f[,2] == [2,1]~, my(p=f[1,1], q=f[2,1]); (((q-1) % p) == 0) && (((q-1) % p^2) != 0);); \\ _Michel Marcus_, Dec 16 2021
%o A349495 (Python)
%o A349495 from sympy import factorint
%o A349495 def ok(n):
%o A349495     if n < 13: return False
%o A349495     f = factorint(n)
%o A349495     sig, p, q = list(f.values()), min(f), max(f)
%o A349495     return sig == [2, 1] and (q-1)%p == 0 and (q-1)%p**2 != 0
%o A349495 print([m for m in range(1422) if ok(m)]) # _Michael S. Branicky_, Dec 16 2021
%Y A349495 Intersection of A054396 and A054753.
%Y A349495 Cf. A002145.
%K A349495 nonn
%O A349495 1,1
%A A349495 _Bernard Schott_, Dec 15 2021
%E A349495 More terms from _Alois P. Heinz_, Dec 15 2021