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
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.
Programs
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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.