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A350245 Numbers p^2*q, p > q odd primes such that q divides p+1.

%I A350245 #21 Dec 28 2021 00:57:16
%S A350245 75,363,867,1183,1587,1805,2523,4205,5043,6627,8427,10443,11767,15123,
%T A350245 17405,20339,20667,23763,26011,30603,31205,34347,38307,39605,48223,
%U A350245 51483,56307,59405,65863,66603,76313,83667,89787,96123,96605,109443,111005,115351,116427
%N A350245 Numbers p^2*q, p > q odd primes such that q divides p+1.
%C A350245 For these terms m, there are precisely 3 groups of order m, so this is a subsequence of A055561.
%C A350245 The 3 groups are C_{p^2*q}, (C_p X C_p) X C_q and (C_p X C_p) : C_q, where C means cyclic groups of the stated order, the symbols X and : mean direct and semidirect products respectively.
%D A350245 Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.
%H A350245 Robert Israel, <a href="/A350245/b350245.txt">Table of n, a(n) for n = 1..10000</a>
%e A350245 75 = 5^2 * 3, 5 and 3 are odd and 3 divides 5+1 = 6, hence 75 is a term.
%e A350245 1183 = 13^2 * 7, 13 and 7 are odd and 7 divides 13+1 = 14, hence 1183 is another term.
%p A350245 N:= 10^6: # for terms <= N
%p A350245 P:= select(isprime, [seq(i,i=3..floor(sqrt(N/3)),2)]):
%p A350245 g:= proc(p) local Q;
%p A350245       Q:= numtheory:-factorset(p+1) minus {2};
%p A350245       select(`<=`, map(q -> p^2*q, Q), N);
%p A350245 end proc:
%p A350245 sort(convert(`union`(op(map(g,P))),list)); # _Robert Israel_, Dec 28 2021
%t A350245 q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {1, 2} && Divisible[p[[2]] + 1, p[[1]]]]; Select[Range[1, 2*10^5, 2], q] (* _Amiram Eldar_, Dec 21 2021 *)
%o A350245 (Python)
%o A350245 from sympy import integer_nthroot, primerange
%o A350245 def aupto(limit):
%o A350245     aset, maxp = set(), integer_nthroot(limit**2, 3)[0]
%o A350245     for p in primerange(3, maxp+1):
%o A350245         pp = p*p
%o A350245         for q in primerange(3, min(p-1, limit//pp)+1):
%o A350245             if (p+1)%q == 0:
%o A350245                 aset.add(pp*q)
%o A350245     return sorted(aset)
%o A350245 print(aupto(120000)) # _Michael S. Branicky_, Dec 21 2021
%Y A350245 Intersection of A054753 and A055561.
%Y A350245 Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115.
%K A350245 nonn
%O A350245 1,1
%A A350245 _Bernard Schott_, Dec 21 2021
%E A350245 More terms from _Amiram Eldar_, Dec 21 2021