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A350115 Numbers p^2*q, p primes such that p^2 divides q-1.

%I A350115 #22 Dec 17 2021 20:40:35
%S A350115 20,52,68,116,148,164,171,212,244,292,333,356,388,404,436,452,548,596,
%T A350115 628,657,692,724,772,788,916,932,964,981,1028,1076,1108,1124,1143,
%U A350115 1172,1252,1268,1348,1396,1412,1467,1492,1556,1588,1604,1629,1636,1684,1732,1791,1796,1828,1844
%N A350115 Numbers p^2*q, p<q primes such that p^2 divides q-1.
%C A350115 For these terms m, there are precisely 5 groups of order m, so this is a subsequence of A054397.
%C A350115 Two of them are abelian: C_{p^2*q}, C_q X C_p X C_p = C_q X (C_p)^2, and the three others that are nonabelian are C_q : (C_p x C_p), and two nonisomorphic semi-direct products C_q : C_p^2. Here C means cyclic groups of the stated order, the symbols X and : mean direct and semidirect products respectively.
%D A350115 Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.
%e A350115 20 = 2^2*5 and 2^2 divides 5-1, hence 20 is a term.
%e A350115 171 = 3^2*19 and 3^2 divides 19-1, hence 171 is another term.
%t A350115 q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {2, 1} && Divisible[p[[2]] - 1, p[[1]]^2]]; Select[Range[2000], q] (* _Amiram Eldar_, Dec 14 2021 *)
%o A350115 (PARI) isok(m) = {my(f=factor(m)); if (f[,2] == [2,1]~, my(p=f[1,1], q=f[2,1]); ((q-1) % p^2) == 0;);} \\ _Michel Marcus_, Dec 14 2021
%o A350115 (Python)
%o A350115 from sympy import integer_nthroot, isprime, primerange
%o A350115 def aupto(limit):
%o A350115     aset, maxp = set(), integer_nthroot(limit, 4)[0]
%o A350115     for p in primerange(1, maxp+1):
%o A350115         m = p**2
%o A350115         for t in range(m, limit//m, m):
%o A350115             if isprime(t+1):
%o A350115                 aset.add(p**2*(t+1))
%o A350115     return sorted(aset)
%o A350115 print(aupto(1844)) # _Michael S. Branicky_, Dec 14 2021
%Y A350115 Other subsequences of A054397: A030078, A079704, A143928.
%Y A350115 Subsequence of A054753.
%K A350115 nonn
%O A350115 1,1
%A A350115 _Bernard Schott_, Dec 14 2021
%E A350115 More terms from _Michel Marcus_, Dec 14 2021