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]

Prog

(Python)
from sympy import primepi, integer_nthroot, mobius
from oeis_sequences.OEISsequences import bisection
def A378767(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x, k)[0]) for k in range(3, x.bit_length()))+sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1)))
    return bisection(f, n, n) # Chai Wah Wu, Jan 07 2026

Crossrefs

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

Sequence in context: A095158 A271422 A379336 * A362148 A391415 A391319

Adjacent sequences: A378764 A378765 A378766 * A378768 A378769 A378770

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 06 2024

Status

approved