A349446 a(n) = A349444(n) + A349445(n).

2, 0, 0, 1, 0, -2, 0, 1, 1, -4, 0, -1, 0, -6, 4, 1, 0, -5, 0, -2, 6, -10, 0, -1, 4, -12, 5, -3, 0, -4, 0, 1, 10, -16, 12, -2, 0, -18, 12, -2, 0, -6, 0, -5, 14, -22, 0, -1, 9, -16, 16, -6, 0, -13, 20, -3, 18, -28, 0, 0, 0, -30, 21, 1, 24, -10, 0, -8, 22, -12, 0, -2, 0, -36, 24, -9, 30, -12, 0, -2, 19, -40, 0, 0, 32

Offset: 1

Author

Antti Karttunen, Nov 18 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A349444, A349445.

Cf. also A349433.

Programs

Mathematica

s[n_] := MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]]; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#]*k[n/#] &, # < n &]; a[n_] := DivisorSum[n,  s[#]*k[n/#] + IntegerExponent[2*#, 2]*kinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2021 *)

PARI

A349446(n) = (A349444(n)+A349445(n)); \\ Needs also code from A349444 and A349445.

Formula

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A349444(d) * A349445(n/d). [As the sequences are Dirichlet inverses of each other]