V0716

Tantus k (in A126706) such that p₂ > q, i.e., A119288(k) < A053669(k).

12, 18, 24, 36, 48, 54, 60, 72, 84, 90, 96, 108, 120, 126, 132, 144, 150, 156, 162, 168, 180, 192, 198, 204, 216, 228, 234, 240, 252, 264, 270, 276, 288, 294, 300, 306, 312, 324, 336, 342, 348, 360, 372, 378, 384, 396, 408, 414, 420, 432, 444, 450, 456, 468, 480, 486, 492, 504, 516, 522, 528, 540, 552, 558, 564, 576, 588, 594, 600, 612, 624, 630, 636, 648, 660

1, 1

Theorem. Tantus k with primorial ϰ implies p₂ < q, i.e., (i.e., V0730 ⊂ V0706).
Proof. Suppose the converse: p₂ > q. This suggests that the second smallest distinct prime factor is larger than the smallest nondivisor prime q. Let S = { p : p ≤ r, r ≤ p₂ }. Since squarefree kernel ϰ = &Prod;S and since q is coprime to k and thus also ϰ by definition, q must exceed r. Clearly, p₂ ∈ S, contradicting supposition. ∎

Table of n, a(n) for n = 1..2^16.

This sequence is { k : Ω(m) > ω(m) > 1, p₂ > q }.

a(1) = 20 since p₂ = 5 and q = 3.
a(2) = 28 since p₂ = 7 and q = 3.

Select[Range[12, 120], Function[{k, f},
  And[PrimeOmega[k] > PrimeNu[k] > 1,
  f[[2, 1]] < SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1],
    ! Divisible[k, #] &]]] @@
  {#, FactorInteger[#]} &] (* Michael De Vlieger, 18 December 2023 *)

Cf. V7 = A126706, V0100 = A053669, V0111 = A002110, V0201 = A001221, V0202 = A001222, V0210 = A055932, V0715, V0701 = A360769, V0730 = A126706 ∩ A055932, V1002 = A119288.

nonn, cnst, clas