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

1,1

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.

Links

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

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.

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A210456 A095158 A271422 * A362148 A062374 A272593

Adjacent sequences: A378754 A378755 A378763 * A378768 A378771 A378772

Keyword

nonn,easy,new

Author

Michael De Vlieger, Dec 06 2024

Status

approved