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%I #36 Sep 08 2022 08:46:26
%S 245,845,847,1445,1859,2023,2527,2645,3179,3703,3757,3971,4693,6137,
%T 6727,6845,6877,8993,9245,9251,9583,10051,10571,10933,11045,12493,
%U 14045,14297,15059,15463,15979,16337,17797,18259,18491,19343,19663,21853,22103,22445,23273
%N Numbers p^2*q, p > q odd primes such that q does not divide p-1, and q does not divide p+1.
%C As odd prime q does not divide p-1 and does not divide also p+1, then q >= 5, so p >= 7.
%C For these terms m, there are precisely 2 groups of order m, so this is a subsequence of A054395.
%C 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 Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.
%e 245 = 7^2 * 5, 5 and 7 are odd primes, 5 does not divide 7-1 = 10 and does not divide 7+1 = 8, hence 245 is a term.
%t q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {1, 2} && ! Or @@ Divisible[p[[2]] + {-1, 1}, p[[1]]]]; Select[Range[1, 24000, 2], q] (* _Amiram Eldar_, Dec 30 2021 *)
%o (Magma) f:=Factorisation; [n:n in [3..24000 ]|#PrimeDivisors(n) eq 2 and f(n)[1][1] lt f(n)[2][1] and f(n)[1][2] eq 1 and f(n)[2][2] eq 2 and (f(n)[2][1]-1) mod f(n)[1][1] ne 0 and (f(n)[2][1]+1) mod f(n)[1][1] ne 0]; // _Marius A. Burtea_, Dec 30 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(1, min(p, limit//pp+1)):
%o if (p-1)%q != 0 and (p+1)%q != 0:
%o aset.add(pp*q)
%o return sorted(aset)
%o print(aupto(24000)) # _Michael S. Branicky_, Dec 30 2021
%o (PARI) isok(m) = my(f=factor(m)); if (f[, 2] == [1, 2]~, my(p=f[2, 1], q=f[1, 1]); ((p-1) % q) && ((p+1) % q)); \\ _Michel Marcus_, Dec 30 2021
%Y Equals A350422 \ A350332.
%Y Subsequence of A051532, A054395, A054753, A060687 and A350322.
%Y Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245.
%K nonn
%O 1,1
%A _Bernard Schott_, Dec 30 2021
%E More terms from _Marius A. Burtea_ and _Hugo Pfoertner_, Dec 30 2021
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