A349495 Numbers p^2*q, p (2,3).

28, 44, 63, 76, 92, 117, 124, 172, 188, 236, 268, 275, 279, 284, 316, 332, 387, 412, 428, 508, 524, 549, 556, 603, 604, 652, 668, 711, 716, 764, 775, 796, 844, 873, 892, 908, 927, 956, 1004, 1025, 1052, 1084, 1132, 1228, 1244, 1251, 1324, 1359, 1388, 1413, 1421

Offset: 1

Author

Bernard Schott, Dec 15 2021

Keywords

• nonn

Comments

For these terms m, there are precisely 4 groups of order m, so this is a subsequence of A054396.

Two of them are abelian: C_{p^2*q}, C_q X C_p X C_p, and the two others that are nonabelian are C_q : (C_p x C_p), and C_q : C_p^2. Note that when p = 2, C_q : (C_p x C_p) ~ D_{p^2*q}. Here C and D mean cyclic and dihedral groups of the stated order, the symbols ~, X and : mean "isomorphic to", direct and semidirect products respectively.

Why (p,q) <> (2,3)? Because there are 5 groups of order 12, and in this particular case, the 5th group is the alternating group A_4 because 2^2*3 = 4!/2 (see Example section in A054397).

Contains 4*r for r in A002145 and r > 3. - Alois P. Heinz, Dec 15 2021

Examples

			28 = 2^2*7, 2 divides 7-1 = 6 and 2^2 does not divide 7-1 = 6, hence 28 is a term.
63 = 3^2*7, 3 divides 7-1 = 6 and 3^2 does not divide 7-1 = 6, hence 63 is another term.
		

References

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

Crossrefs

Intersection of A054396 and A054753.

Cf. A002145.

Programs

Mathematica

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {2, 1} && IntegerExponent[p[[2]] - 1, p[[1]]] == 1]; Select[Range[28, 1500], q]  (* Amiram Eldar, Dec 16 2021 *)

PARI

isok(m) = if (m==12, return(0)); my(f=factor(m)); if (f[,2] == [2,1]~, my(p=f[1,1], q=f[2,1]); (((q-1) % p) == 0) && (((q-1) % p^2) != 0);); \\ Michel Marcus, Dec 16 2021

Python

from sympy import factorint
def ok(n):
    if n < 13: return False
    f = factorint(n)
    sig, p, q = list(f.values()), min(f), max(f)
    return sig == [2, 1] and (q-1)%p == 0 and (q-1)%p**2 != 0
print([m for m in range(1422) if ok(m)]) # Michael S. Branicky, Dec 16 2021

Extensions

More terms from Alois P. Heinz, Dec 15 2021