A369150 Numbers k neither squarefree nor prime powers such that A053669(k) < k/rad(k) < A119288(k) that are not odd numbers of the form lpf(k)*rad(k), where lpf(k) = A020639(k) and rad(k) = A007947(k).

40, 56, 88, 104, 136, 152, 176, 184, 208, 232, 248, 272, 280, 296, 297, 304, 328, 344, 351, 368, 376, 424, 440, 459, 464, 472, 488, 496, 513, 520, 536, 544, 568, 584, 592, 608, 616, 621, 632, 656, 664, 680, 688, 712, 728, 736, 752, 760, 776, 783, 808, 824, 837

Offset: 1

Author

Michael De Vlieger, Jan 20 2024

Keywords

• nonn

Comments

Numbers k neither squarefree nor prime powers such that the smallest nondivisor prime q < k/rad(k) < p, the second smallest prime factor of k where k/rad(k) != lpf(k).

Even k implies A053669(k) = 3, odd k implies A053669(k) = 2.

Sequence does not contain k divisible by 6; sequence does not meet A055932.

Proper subset of A367455.

Examples

			a(1) = 40 = 2^3 * 5, since 3 < 4 < 5 and 4 != 2.
a(2) = 56 = 2^3 * 7, since 3 < 4 < 7 and 4 != 2.
a(7) = 176 = 2^4 * 11, since 3 < 8 < 11 and 8 != 2.
a(15) = 297 = 3^3 * 11, since 2 < 9 < 11 and 9 != 3.
a(248) = 3625 = 5^3 * 29, since 2 < 25 < 29 and 25 != 5, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007947, A020639, A053669, A119288, A126706, A364997, A366460, A366825, A367455.

Programs

Mathematica

s = Select[Range[1000], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
Select[s,
  And[#3 < #1 < #2, #1 != #4] & @@
  {#1/(Times @@ #2), #2[[2]], #3, First[#2]} & @@
  {#, FactorInteger[#][[All, 1]],
    If[OddQ[#], 2, q = 3; While[Divisible[#, q], q = NextPrime[q]]; q]} &]

Formula

This sequence is { A364997 \ A366460 } = { A364997 \ A366825 }.