A350245 Numbers p^2*q, p > q odd primes such that q divides p+1.

75, 363, 867, 1183, 1587, 1805, 2523, 4205, 5043, 6627, 8427, 10443, 11767, 15123, 17405, 20339, 20667, 23763, 26011, 30603, 31205, 34347, 38307, 39605, 48223, 51483, 56307, 59405, 65863, 66603, 76313, 83667, 89787, 96123, 96605, 109443, 111005, 115351, 116427

Offset: 1

Author

Bernard Schott, Dec 21 2021

Keywords

• nonn

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.

Examples

			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.
		

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

Crossrefs

Intersection of A054753 and A055561.

Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115.

Programs

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 *)

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

Extensions

More terms from Amiram Eldar, Dec 21 2021