V0715

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

20, 28, 40, 44, 45, 50, 52, 56, 63, 68, 75, 76, 80, 88, 92, 98, 99, 100, 104, 112, 116, 117, 124, 135, 136, 140, 147, 148, 152, 153, 160, 164, 171, 172, 175, 176, 184, 188, 189, 196, 200, 207, 208, 212, 220, 224, 225, 232, 236, 242, 244, 245, 248, 250, 260, 261, 268, 272, 275, 279, 280, 284, 292, 296, 297, 304, 308, 315, 316, 320, 325, 328, 332, 333, 338, 340

1, 1

Theorem. Odd tantus k implies p₂ > q, i.e., (i.e., A360769 ⊂ V0705).
Proof. If k is odd, q = 2, and since primes must be divisors or not and 2 is the smallest prime, it follows that p₂ > q.

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, V0716, V0701 = A360769, V0730 = A126706 ∩ A055932, V1002 = A119288.

nonn, cnst, clas