login
Numbers p^2*q, p<q primes such that p^2 divides q-1.
7

%I #22 Dec 17 2021 20:40:35

%S 20,52,68,116,148,164,171,212,244,292,333,356,388,404,436,452,548,596,

%T 628,657,692,724,772,788,916,932,964,981,1028,1076,1108,1124,1143,

%U 1172,1252,1268,1348,1396,1412,1467,1492,1556,1588,1604,1629,1636,1684,1732,1791,1796,1828,1844

%N Numbers p^2*q, p<q primes such that p^2 divides q-1.

%C For these terms m, there are precisely 5 groups of order m, so this is a subsequence of A054397.

%C 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.

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

%e 20 = 2^2*5 and 2^2 divides 5-1, hence 20 is a term.

%e 171 = 3^2*19 and 3^2 divides 19-1, hence 171 is another term.

%t 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 *)

%o (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

%o (Python)

%o from sympy import integer_nthroot, isprime, primerange

%o def aupto(limit):

%o aset, maxp = set(), integer_nthroot(limit, 4)[0]

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

%o m = p**2

%o for t in range(m, limit//m, m):

%o if isprime(t+1):

%o aset.add(p**2*(t+1))

%o return sorted(aset)

%o print(aupto(1844)) # _Michael S. Branicky_, Dec 14 2021

%Y Other subsequences of A054397: A030078, A079704, A143928.

%Y Subsequence of A054753.

%K nonn

%O 1,1

%A _Bernard Schott_, Dec 14 2021

%E More terms from _Michel Marcus_, Dec 14 2021