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.
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
KEYWORD
nonn
AUTHOR
Bernard Schott, Dec 14 2021
EXTENSIONS
More terms from Michel Marcus, Dec 14 2021
STATUS
approved