A375286 a(n) = f(1) + f(2) + ... + f(n), where f(n) = (-2)^Omega(n) = A165872(n).

1, -1, -3, 1, -1, 3, 1, -7, -3, 1, -1, -9, -11, -7, -3, 13, 11, 3, 1, -7, -3, 1, -1, 15, 19, 23, 15, 7, 5, -3, -5, -37, -33, -29, -25, -9, -11, -7, -3, 13, 11, 3, 1, -7, -15, -11, -13, -45, -41, -49, -45, -53, -55, -39, -35, -19, -15, -11, -13, 3, 1, 5, -3, 61

Offset: 1

Author

Charles R Greathouse IV, Aug 09 2024

Keywords

• sign
• look

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Daniel R. Johnston, Nicol Leong, and Sebastian Tudzi, New bounds and progress towards a conjecture on the summatory function of (-2)^Omega(n), arXiv:2408.04143 [math.NT], 2024.
• Michael J. Mossinghoff and Timothy S. Trudgian, Oscillations in weighted arithmetic sums, arXiv:2007.14537 [math.NT], 2020.
• Zhi-Wei Sun, On a pair of zeta functions, arXiv:1204.6689 [math.NT], 2012.

Crossrefs

Partial sums of A165872.

Cf. A001222, A002321, A212496, A061142, A069205.

Programs

Maple

a:= proc(n) option remember; `if`(n<1, 0,
      a(n-1)+(-2)^numtheory[bigomega](n))
    end:
seq(a(n), n=1..64);  # Alois P. Heinz, Apr 25 2025

PARI

s=0; vector(60,n,s+=(-2)^bigomega(n))

Formula

Johnston, Leong, & Tudzi prove that |a(n)| < 2260n. Sun conjectures that |a(n)| < n for n >= 3078. Mossinghoff & Trudgian verify this to 2.5 * 10^14.

Because of powers of two, |a(n)| >= n/2 infinitely often.