A350638 Numbers of the form p^2*q, with odd primes p > q, such that q divides p-1.

147, 507, 605, 1083, 2883, 4107, 4805, 5547, 5819, 5887, 8405, 11163, 12943, 13467, 15987, 18605, 18723, 25205, 28227, 31827, 35287, 35643, 36517, 48387, 49379, 50807, 51005, 57963, 68403, 73947, 79707, 81133, 85805, 87131, 89383, 98283, 100949, 111747, 112903

Offset: 1

Author

Bernard Schott, Jan 10 2022

Keywords

• nonn

Comments

For these terms m there are precisely (q+9)/2 groups of order m.

Only two of these groups are abelian: C_{p^2*q} and (C_p X C_p) X C_q. The (q+5)/2 groups that are nonabelian are C_{p^2} : C_q and the (q+3)/2 semidirect products of the form (C_p X C_p) : C_q that are not isomorphic, where C means cyclic groups of the stated order, the symbols X and : mean direct and semidirect products respectively.

Examples

			147 = 7^2 * 3, 3 and 7 are odd primes, 3 divides 7-1 = 6, hence 147 is a term.
		

References

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

Links

• N. S. Wedd, Groups of orders 147.

Crossrefs

Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245, A350332, A350421, A350422.

Programs

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, 120000, 2], q] (* Amiram Eldar, Jan 11 2022 *)

PARI

isok(m) = if (m%2, my(f=factor(m)); if (f[, 2] == [1, 2]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p) == 0)); \\ Michel Marcus, Jan 11 2022

Python

from sympy import integer_nthroot, primerange
def aupto(limit):
    aset, maxp = set(), integer_nthroot(limit**2, 3)[0]
    for p in primerange(5, maxp+1):
        pp = p*p
        for q in primerange(3, min(p, limit//pp+1)):
            if (p-1)%q == 0:
                aset.add(pp*q)
    return sorted(aset)
print(aupto(113000)) # Michael S. Branicky, Jan 10 2022

Extensions

More terms from Michael S. Branicky, Jan 10 2022