A346483 Sum of A005171 (characteristic function of nonprimes) and its Dirichlet inverse.

2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 2, 0

Offset: 1

Author

Mats Granvik and Antti Karttunen, Aug 17 2021

Keywords

• sign

Comments

The first negative term is a(192) = -1.

Positions of nonzero terms are given by A033987, except for positions n = 256, 512, 6561, 16384, 19683, 32768, 390625, 1048576, ..., at which a(n) = 0 also.

Links

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

Crossrefs

Cf. A005171, A010051, A033987, A346482.

Programs

Mathematica

nn = 87; b = Table[If[PrimeQ[n], 1, 0], {n, nn}]; a = 1 - b; A = Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]; B = Inverse[A]; S = A[[Range[nn]]] + B[[Range[nn]]]; S[[All, 1]]

PARI

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA005171(n) = (1-isprime(n));
v346482 = DirInverseCorrect(vector(up_to,n,A005171(n)));
A346482(n) = v346482[n];
A346483(n) = (A005171(n)+A346482(n));

Formula

a(n) = A005171(n) + A346482(n).

For n > 1, a(n) = -Sum_{d|n, 1A005171(d) * A346482(n/d).