OFFSET
1,1
COMMENTS
For these terms m, there are precisely 3 groups of order m, so this is a subsequence of A055561.
The 3 groups are C_{p^2*q}, (C_p X C_p) X C_q and (C_p X C_p) : C_q, where 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
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
75 = 5^2 * 3, 5 and 3 are odd and 3 divides 5+1 = 6, hence 75 is a term.
1183 = 13^2 * 7, 13 and 7 are odd and 7 divides 13+1 = 14, hence 1183 is another term.
MAPLE
N:= 10^6: # for terms <= N
P:= select(isprime, [seq(i, i=3..floor(sqrt(N/3)), 2)]):
g:= proc(p) local Q;
Q:= numtheory:-factorset(p+1) minus {2};
select(`<=`, map(q -> p^2*q, Q), N);
end proc:
sort(convert(`union`(op(map(g, P))), list)); # Robert Israel, Dec 28 2021
MATHEMATICA
q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {1, 2} && Divisible[p[[2]] + 1, p[[1]]]]; Select[Range[1, 2*10^5, 2], q] (* Amiram Eldar, Dec 21 2021 *)
PROG
(Python)
from sympy import integer_nthroot, primerange
def aupto(limit):
aset, maxp = set(), integer_nthroot(limit**2, 3)[0]
for p in primerange(3, maxp+1):
pp = p*p
for q in primerange(3, min(p-1, limit//pp)+1):
if (p+1)%q == 0:
aset.add(pp*q)
return sorted(aset)
print(aupto(120000)) # Michael S. Branicky, Dec 21 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Dec 21 2021
EXTENSIONS
More terms from Amiram Eldar, Dec 21 2021
STATUS
approved