A309082 a(n) = n - floor(n/2^3) + floor(n/3^3) - floor(n/4^3) + ...

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 43, 44, 45, 46, 47, 48, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 57, 58, 59, 60, 61, 62, 63, 64, 64, 65, 66, 67

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Formula

G.f.: (1/(1 - x)) * Sum_{k>=1} (-1)^(k+1) * x^(k^3)/(1 - x^(k^3)).

a(n) ~ 3*zeta(3)*n/4. - Vaclav Kotesovec, Oct 12 2019

From Ridouane Oudra, Nov 01 2025: (Start)

a(n) = Sum_{k=1..floor(n^(1/3))} (-1)^(k+1)*floor(n/k^3).

a(n) = Sum_{k=1..n} (floor((n/k)^(1/3)) mod 2).

a(n) = Sum_{n/8 < k <= n} A061704(k) - Sum_{1 <= k <= n/8} A061704(k).

a(n) = Sum_{k=1..n} A390259(k).

a(n) = A013937(n) - 2*A013937(floor(n/8)). (End)

Mathematica

Table[Sum[(-1)^(k + 1) Floor[n/k^3], {k, 1, n}], {n, 1, 75}]
(* Alternative: *)
nmax = 75; CoefficientList[Series[1/(1 - x) Sum[(-1)^(k + 1) x^(k^3)/(1 - x^(k^3)), {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
(* Alternative: *)
Table[Sum[Boole[IntegerQ[d^(1/3)] && OddQ[d]], {d, Divisors[n]}] - Sum[Boole[IntegerQ[d^(1/3)] && EvenQ[d]], {d, Divisors[n]}], {n, 1, 75}] // Accumulate

Prog

(Python)
from sympy import integer_nthroot
def A309082(n):
    c, j, w = 0, 1, 0
    while (j2:=j**3) <= n:
        k = n//j2
        m = integer_nthroot(n//k, 3)[0]
        c += (-w+(w:=m&1))*k
        j = m+1
    return c # Chai Wah Wu, Jun 23 2026

Crossrefs

Cf. A013937, A059851, A061704, A309081, A309083, A390259.

Sequence in context: A060431 A228298 A368918 * A124808 A238401 A337978

Adjacent sequences: A309079 A309080 A309081 * A309083 A309084 A309085

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 11 2019

Status

approved