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

%I A375286 #32 Apr 25 2025 08:20:19
%S A375286 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,
%T A375286 23,15,7,5,-3,-5,-37,-33,-29,-25,-9,-11,-7,-3,13,11,3,1,-7,-15,-11,
%U A375286 -13,-45,-41,-49,-45,-53,-55,-39,-35,-19,-15,-11,-13,3,1,5,-3,61
%N A375286 a(n) = f(1) + f(2) + ... + f(n), where f(n) = (-2)^Omega(n) = A165872(n).
%H A375286 Alois P. Heinz, <a href="/A375286/b375286.txt">Table of n, a(n) for n = 1..20000</a>
%H A375286 Daniel R. Johnston, Nicol Leong, and Sebastian Tudzi, <a href="https://arxiv.org/abs/2408.04143">New bounds and progress towards a conjecture on the summatory function of (-2)^Omega(n)</a>, arXiv:2408.04143 [math.NT], 2024.
%H A375286 Michael J. Mossinghoff and Timothy S. Trudgian, <a href="https://arxiv.org/abs/2007.14537">Oscillations in weighted arithmetic sums</a>, arXiv:2007.14537 [math.NT], 2020.
%H A375286 Zhi-Wei Sun, <a href="https://arxiv.org/abs/1204.6689">On a pair of zeta functions</a>, arXiv:1204.6689 [math.NT], 2012.
%F A375286 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.
%F A375286 Because of powers of two, |a(n)| >= n/2 infinitely often.
%p A375286 a:= proc(n) option remember; `if`(n<1, 0,
%p A375286       a(n-1)+(-2)^numtheory[bigomega](n))
%p A375286     end:
%p A375286 seq(a(n), n=1..64);  # _Alois P. Heinz_, Apr 25 2025
%o A375286 (PARI) s=0; vector(60,n,s+=(-2)^bigomega(n))
%Y A375286 Partial sums of A165872.
%Y A375286 Cf. A001222, A002321, A212496, A061142, A069205.
%K A375286 sign,look
%O A375286 1,3
%A A375286 _Charles R Greathouse IV_, Aug 09 2024