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

Offset: 1

Author

Bernard Schott, Dec 14 2021

Keywords

• nonn

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.

Examples

			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.
		

References

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

Crossrefs

Other subsequences of A054397: A030078, A079704, A143928.

Subsequence of A054753.

Programs

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 *)

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

Extensions

More terms from Michel Marcus, Dec 14 2021