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Numbers p^2*q, p > q odd primes such that q divides p+1.
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%I #21 Dec 28 2021 00:57:16

%S 75,363,867,1183,1587,1805,2523,4205,5043,6627,8427,10443,11767,15123,

%T 17405,20339,20667,23763,26011,30603,31205,34347,38307,39605,48223,

%U 51483,56307,59405,65863,66603,76313,83667,89787,96123,96605,109443,111005,115351,116427

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

%C For these terms m, there are precisely 3 groups of order m, so this is a subsequence of A055561.

%C 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 Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.

%H Robert Israel, <a href="/A350245/b350245.txt">Table of n, a(n) for n = 1..10000</a>

%e 75 = 5^2 * 3, 5 and 3 are odd and 3 divides 5+1 = 6, hence 75 is a term.

%e 1183 = 13^2 * 7, 13 and 7 are odd and 7 divides 13+1 = 14, hence 1183 is another term.

%p N:= 10^6: # for terms <= N

%p P:= select(isprime, [seq(i,i=3..floor(sqrt(N/3)),2)]):

%p g:= proc(p) local Q;

%p Q:= numtheory:-factorset(p+1) minus {2};

%p select(`<=`, map(q -> p^2*q, Q), N);

%p end proc:

%p sort(convert(`union`(op(map(g,P))),list)); # _Robert Israel_, Dec 28 2021

%t 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 (Python)

%o from sympy import integer_nthroot, primerange

%o def aupto(limit):

%o aset, maxp = set(), integer_nthroot(limit**2, 3)[0]

%o for p in primerange(3, maxp+1):

%o pp = p*p

%o for q in primerange(3, min(p-1, limit//pp)+1):

%o if (p+1)%q == 0:

%o aset.add(pp*q)

%o return sorted(aset)

%o print(aupto(120000)) # _Michael S. Branicky_, Dec 21 2021

%Y Intersection of A054753 and A055561.

%Y Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115.

%K nonn

%O 1,1

%A _Bernard Schott_, Dec 21 2021

%E More terms from _Amiram Eldar_, Dec 21 2021