A378767 Numbers k that are not prime powers which are divisible by a cube greater than 1.

24, 40, 48, 54, 56, 72, 80, 88, 96, 104, 108, 112, 120, 135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 248, 250, 264, 270, 272, 280, 288, 296, 297, 304, 312, 320, 324, 328, 336, 344, 351, 352, 360, 368, 375, 376, 378, 384

Offset: 1

Author

Michael De Vlieger, Dec 06 2024

Keywords

• nonn
• easy

Comments

Products m = j*k such that omega(k) = omega(m) > omega(j), where rad(j) | k but j does not divide k, with rad = A007947 and omega = A001221.

Proper subset of A126706.

This sequence is distinct from A362148, since this sequence also contains 216, 432, etc.

Examples

			Prime decomposition of select a(n) = m, showing m = j*k:
a(1) = 24 = 2^3 * 3 = 4 * 6.
a(2) = 40 = 2^3 * 5 = 4 * 10.
a(3) = 48 = 2^4 * 3 = 8 * 6.
a(4) = 54 = 2 * 3^3 = 9 * 6.
a(5) = 56 = 2^3 * 7 = 4 * 14.
a(6) = 72 = 2^3 * 3^2 = 4 * 18.
a(9) = 96 = 2^5 * 3 = 8 * 12 = 16 * 6.
a(130) = 864 = 2^5 * 3^2 = 8 * 108 = 9 * 96 = 16 * 54, etc.
		

Links

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

Crossrefs

Cf. A024619, A046099, A126706, A362148, A372695.

Programs

Mathematica

Select[Select[Range[2^10], AnyTrue[FactorInteger[#][[All, -1]], # > 2 &] &], Not@*PrimePowerQ]

Formula

{a(n)} = { k : omega(k) > 1, there exists p | k such that p^3 | k }.

Intersection of A046099 and A024619.

Union of A362148 and A372695.