A054396 -id:A054396 - cp's OEIS frontend

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

Bernard Schott, Dec 15 2021

nonn

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

			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.

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

Intersection of A054396 and A054753.

Cf. A002145.

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 *)

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

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

More terms from Alois P. Heinz, Dec 15 2021

A296022 Numbers n such that there are precisely 2 groups of orders n, n + 1 and n + 2.

Original entry on oeis.org

201, 205, 325, 1045, 1653, 1857, 1965, 2041, 2301, 2305, 2605, 2637, 2653, 2853, 2973, 3241, 3445, 3505, 3721, 3757, 4173, 4405, 4585, 4693, 5005, 5217, 5241, 5341, 5685, 5757, 5853, 6685, 6745, 7285, 8005, 8845, 9325, 9441, 9777, 10201, 10293, 10417, 10833

Offset: 1

Muniru A Asiru, Dec 03 2017

nonn

Equivalently, lower member of consecutive terms of A295230.

			n = 201 -> A000001(201) = A000001(202) = A000001(203) = 2.
n = 205 -> A000001(205) = A000001(206) = A000001(207) = 2.
n = 1965 -> A000001(1965) = A000001(1966) = A000001(1967) = 2.

Muniru A Asiru, Table of n, a(n) for n = 1..2583
H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
Gordon Royle, Numbers of Small Groups
Index entries for sequences related to groups

Cf. A000001, A054396. Subsequence of A295230.

A296022 := Filtered([1..2013], n -> [NumberSmallGroups(n), NumberSmallGroups(n+1), NumberSmallGroups(n+2)]=[2, 2, 2]);

with(GroupTheory): with(numtheory):
for n from 1 to 10^4 do if [NumGroups(n),NumGroups(n+1),NumGroups(n+2)]=[2,2,2]  then print(n); fi; od;

cnt = FiniteGroupCount; Select[Range[10^4], cnt[#] == cnt[#+1] == cnt[#+2] == 2&] (* Jean-François Alcover, Dec 08 2017 *)

Sequence is { n | A000001(n) = 2, A000001(n+1) = 2, A000001(n+2) = 2 }.

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

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