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A350332 Numbers p^2*q, p < q odd primes such that p does not divide q-1.

%I A350332 #18 Dec 25 2021 15:00:37
%S A350332 45,99,153,175,207,261,325,369,423,425,475,477,531,539,575,637,639,
%T A350332 725,747,801,833,909,925,931,963,1017,1075,1127,1175,1179,1233,1325,
%U A350332 1341,1475,1503,1519,1557,1573,1611,1675,1719,1773,1813,1825,1975,2009,2043,2057
%N A350332 Numbers p^2*q, p < q odd primes such that p does not divide q-1.
%C A350332 For these terms m, there are precisely 2 groups of order m, so this is a subsequence of A054395.
%C A350332 The 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 A350332 Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.
%e A350332 99 = 3^2 * 11, 3 and 11 are odd and 3 does not divide 11-1 = 10, hence 99 is a term.
%e A350332 175 = 5^2 * 7, 5 and 7 are odd and 5 does not divide 7-1 = 6, hence 115 is another term.
%t A350332 q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {2, 1} && ! Divisible[p[[2]] - 1, p[[1]]]]; Select[Range[2000], q] (* _Amiram Eldar_, Dec 25 2021 *)
%o A350332 (Python)
%o A350332 from sympy import integer_nthroot, primerange
%o A350332 def aupto(limit):
%o A350332     aset, maxp = set(), integer_nthroot(limit, 3)[0]
%o A350332     for p in primerange(3, maxp+1):
%o A350332         pp = p*p
%o A350332         for q in primerange(p+1, limit//pp+1):
%o A350332             if (q-1)%p != 0:
%o A350332                 aset.add(pp*q)
%o A350332     return sorted(aset)
%o A350332 print(aupto(2060)) # _Michael S. Branicky_, Dec 25 2021
%o A350332 (PARI) isok(m) = my(f=factor(m)); if (f[, 2] == [2, 1]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p)); \\ _Michel Marcus_, Dec 25 2021
%Y A350332 Subsequence of A051532, A054395, A054753 and of A060687.
%Y A350332 Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245.
%K A350332 nonn
%O A350332 1,1
%A A350332 _Bernard Schott_, Dec 25 2021
%E A350332 More terms from _Michael S. Branicky_, Dec 25 2021