A358038 Partial sums of the cubefree numbers.

1, 3, 6, 10, 15, 21, 28, 37, 47, 58, 70, 83, 97, 112, 129, 147, 166, 186, 207, 229, 252, 277, 303, 331, 360, 390, 421, 454, 488, 523, 559, 596, 634, 673, 714, 756, 799, 843, 888, 934, 981, 1030, 1080, 1131, 1183, 1236, 1291, 1348, 1406, 1465, 1525, 1586, 1648, 1711

Offset

1,2

Comments

First differs from A025706 and A025730 at n = 14.

The analogous sequence with squarefree numbers is A173143.

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.

Formula

a(n) = Sum_{k=1..n} A004709(k).

a(n) = (zeta(3)/2)*n^2 + O(n^(3/2+eps)) (Weiyi, 2004).

Mathematica

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

Prog

(Python)
from math import comb
from sympy import mobius, integer_nthroot
def A358038(n):
    def f(x): return int(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 sum(mobius(k)*(s:=k**3)*comb(m//s+1, 2) for k in range(1, integer_nthroot(m, 3)[0]+1)) # Chai Wah Wu, Oct 20 2025

Crossrefs

Cf. A002117, A004709, A173143.

Cf. A025706, A025730.

Sequence in context: A025737 A120721 A122056 * A025706 A025730 A384328

Adjacent sequences: A358035 A358036 A358037 * A358039 A358040 A358041

Keyword

nonn

Author

Amiram Eldar, Oct 29 2022

Extensions

Corrected by Amiram Eldar, Jan 02 2024

Status

approved