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 A350115 Numbers p^2*q, p
 20, 52, 68, 116, 148, 164, 171, 212, 244, 292, 333, 356, 388, 404, 436, 452, 548, 596, 628, 657, 692, 724, 772, 788, 916, 932, 964, 981, 1028, 1076, 1108, 1124, 1143, 1172, 1252, 1268, 1348, 1396, 1412, 1467, 1492, 1556, 1588, 1604, 1629, 1636, 1684, 1732, 1791, 1796, 1828, 1844 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For these terms m, there are precisely 5 groups of order m, so this is a subsequence of A054397. 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. REFERENCES Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004. LINKS Table of n, a(n) for n=1..52. EXAMPLE 20 = 2^2*5 and 2^2 divides 5-1, hence 20 is a term. 171 = 3^2*19 and 3^2 divides 19-1, hence 171 is another term. MATHEMATICA 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 *) PROG (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 (Python) from sympy import integer_nthroot, isprime, primerange def aupto(limit): aset, maxp = set(), integer_nthroot(limit, 4)[0] for p in primerange(1, maxp+1): m = p**2 for t in range(m, limit//m, m): if isprime(t+1): aset.add(p**2*(t+1)) return sorted(aset) print(aupto(1844)) # Michael S. Branicky, Dec 14 2021 CROSSREFS Other subsequences of A054397: A030078, A079704, A143928. Subsequence of A054753. Sequence in context: A220040 A128905 A276135 * A211143 A183047 A209982 Adjacent sequences: A350112 A350113 A350114 * A350116 A350117 A350118 KEYWORD nonn AUTHOR Bernard Schott, Dec 14 2021 EXTENSIONS More terms from Michel Marcus, Dec 14 2021 STATUS approved

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Last modified September 15 11:45 EDT 2024. Contains 375938 sequences. (Running on oeis4.)