A350332 Numbers p^2*q, p < q odd primes such that p does not divide q-1.

45, 99, 153, 175, 207, 261, 325, 369, 423, 425, 475, 477, 531, 539, 575, 637, 639, 725, 747, 801, 833, 909, 925, 931, 963, 1017, 1075, 1127, 1175, 1179, 1233, 1325, 1341, 1475, 1503, 1519, 1557, 1573, 1611, 1675, 1719, 1773, 1813, 1825, 1975, 2009, 2043, 2057

Offset: 1

Author

Bernard Schott, Dec 25 2021

Keywords

• nonn

Comments

For these terms m, there are precisely 2 groups of order m, so this is a subsequence of A054395.

The 2 groups are abelian; they are C_{p^2*q} and (C_p X C_p) X C_q, where C means cyclic groups of the stated order and the symbol X means direct product.

Examples

			99 = 3^2 * 11, 3 and 11 are odd and 3 does not divide 11-1 = 10, hence 99 is a term.
175 = 5^2 * 7, 5 and 7 are odd and 5 does not divide 7-1 = 6, hence 115 is another term.
		

References

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

Crossrefs

Subsequence of A051532, A054395, A054753 and of A060687.

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

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]]]]; Select[Range[2000], q] (* Amiram Eldar, Dec 25 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)); \\ Michel Marcus, Dec 25 2021

Python

from sympy import integer_nthroot, primerange
def aupto(limit):
    aset, maxp = set(), integer_nthroot(limit, 3)[0]
    for p in primerange(3, maxp+1):
        pp = p*p
        for q in primerange(p+1, limit//pp+1):
            if (q-1)%p != 0:
                aset.add(pp*q)
    return sorted(aset)
print(aupto(2060)) # Michael S. Branicky, Dec 25 2021

Extensions

More terms from Michael S. Branicky, Dec 25 2021