A350422 Numbers of the form m = p^2*q for which there exist exactly 2 groups of order m.

45, 99, 153, 175, 207, 245, 261, 325, 369, 423, 425, 475, 477, 531, 539, 575, 637, 639, 725, 747, 801, 833, 845, 847, 909, 925, 931, 963, 1017, 1075, 1127, 1175, 1179, 1233, 1325, 1341, 1445, 1475, 1503, 1519, 1557, 1573, 1611, 1675, 1719, 1773, 1813, 1825, 1859, 1975, 2009

Offset: 1

Author

Bernard Schott, Jan 03 2022

Keywords

• nonn

Comments

Terms come from the union of terms of the form p^2*q with p < q in A350332 and terms of the same form with p > q in A350421, with p, q odd primes.

All terms are odd.

These 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

			With p < q: 175 = 5^2 * 7, 5 and 7 are odd primes and 5 does not divide 7-1 = 6, hence 175 is a term (see A350332).
With p > q: 245 = 7^2 * 5, 5 and 7 are odd primes, 5 does not divide 7-1 = 6 and does not divide 7+1 = 8, hence 245 is a term (see A350421).
		

References

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

Crossrefs

Disjoint union of A350332 (pA350421 (p>q).

Intersection of A054395 and A054753.

Subsequence of A051532, A060687 and A350322.

Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245, A350638.

Programs

Mathematica

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (e == {1, 2} && ! Or @@ Divisible[p[[2]] + {-1, 1}, p[[1]]]) || (e == {2, 1} && ! Divisible[p[[2]] - 1, p[[1]]])]; Select[Range[1, 2000, 2], q] (* Amiram Eldar, Jan 03 2022 *)

PARI

isoka(f) = if (f[, 2] == [2, 1]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p)); \\ A350332
isokb(f) = if (f[, 2] == [1, 2]~, my(p=f[2, 1], q=f[1, 1]); ((p-1) % q) && ((p+1) % q)); \\ A350421
isok(m) = my(f=factor(m)); isoka(f) || isokb(f); \\ Michel Marcus, Jan 09 2022