A358040 a(n) is the number of divisors of the n-th cubefree number.

1, 2, 2, 3, 2, 4, 2, 3, 4, 2, 6, 2, 4, 4, 2, 6, 2, 6, 4, 4, 2, 3, 4, 6, 2, 8, 2, 4, 4, 4, 9, 2, 4, 4, 2, 8, 2, 6, 6, 4, 2, 3, 6, 4, 6, 2, 4, 4, 4, 2, 12, 2, 4, 6, 4, 8, 2, 6, 4, 8, 2, 2, 4, 6, 6, 4, 8, 2, 4, 2, 12, 4, 4, 4, 2, 12, 4, 6, 4, 4, 4, 2, 6, 6, 9, 2

Offset: 1

Author

Amiram Eldar, Oct 29 2022

Keywords

• nonn

Comments

The analogous sequence with squarefree numbers is A072048.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Zhu Weiyi, On the cube free number sequences, Smarandache Notions J., Vol. 14 (2004), pp. 199-202.

Crossrefs

Cf. A000005, A001620 (gamma), A004709, A072048, A073002 (-zeta'(2)), A147533 (2*gamma-1), A358039.

Programs

Mathematica

DivisorSigma[0, Select[Range[100], Max[FactorInteger[#][[;;, 2]]] < 3 &]]

Python

from sympy import mobius, integer_nthroot, divisor_count
def A358040(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return divisor_count(m) # Chai Wah Wu, Aug 06 2024

Formula

a(n) = A000005(A004709(n)).

Sum_{k=1..n} a(k) = (36*c_1/Pi^4) * n * (log(n) + (2*gamma - 1) - 24*zeta'(2)/Pi^2 - 4*c_2) + O(n^(1/2 + eps)), where c_1 = Product_{p prime} ((p^2+2*p+3)/(p+1)^2) = 1.58095136661854869148023... and c_2 = Sum_{p prime} p*log(p)/((p+1)*(p^2+2*p+3)) = 0.229224... (Weiyi, 2004).