A349134 Dirichlet inverse of Kimberling's paraphrases, A003602.

1, -1, -2, 0, -3, 2, -4, 0, -1, 3, -6, 0, -7, 4, 4, 0, -9, 1, -10, 0, 5, 6, -12, 0, -4, 7, -2, 0, -15, -4, -16, 0, 7, 9, 6, 0, -19, 10, 8, 0, -21, -5, -22, 0, 3, 12, -24, 0, -9, 4, 10, 0, -27, 2, 8, 0, 11, 15, -30, 0, -31, 16, 4, 0, 9, -7, -34, 0, 13, -6, -36, 0, -37, 19, 8, 0, 9, -8, -40, 0, -4, 21, -42, 0, 11, 22

Offset: 1

Author

Antti Karttunen, Nov 13 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A003602, A349135, A349136.

Cf. also A323881, A349125.

Programs

Mathematica

k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#]*k[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

PARI

up_to = 16384;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003602(n) = (1+(n>>valuation(n,2)))/2;
v349134 = DirInverseCorrect(vector(up_to,n,A003602(n)));
A349134(n) = v349134[n];

Formula

a(1) = 1; a(n) = -Sum_{d|n, d < n} A003602(n/d) * a(d).

a(n) = A349135(n) - A003602(n).