A375072 Biquadratefree numbers (A046100) that are not cubefree (A004709).

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594, 600

Offset: 1

Author

Amiram Eldar, Jul 29 2024

Keywords

• nonn
• easy

Comments

Subsequence of A176297 and first differs from it at n = 41: A176297(41) = 432 = 2^4 * 3^3 is not a term of this sequence.

Numbers whose prime factorization has least one exponent that equals 3 and no higher exponent.

Numbers k such that A051903(k) = 3.

The asymptotic density of this sequence is 1/zeta(4) - 1/zeta(3) = A215267 - A088453 = 0.0920310303408826983406... .

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Intersection of A046100 and A176297.

Cf. A004709, A051903, A067259.

Cf. A002117, A013662, A088453, A215267.

Programs

Mathematica

Select[Range[600], Max[FactorInteger[#][[;; , 2]]] == 3 &]

PARI

is(k) = k > 1 && vecmax(factor(k)[,2]) == 3;

Python

from sympy import mobius, integer_nthroot
def A375072(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**4-x//k**3) for k in range(1, integer_nthroot(x,4)[0]+1))+sum(mobius(k)*(x//k**3) for k in range(integer_nthroot(x,4)[0]+1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024