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

1,1

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

Formula

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

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

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]]

Prog

(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(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
A005171(n) = (1-isprime(n));
v346482 = DirInverseCorrect(vector(up_to, n, A005171(n)));
A346482(n) = v346482[n];
A346483(n) = (A005171(n)+A346482(n));

Crossrefs

Cf. A005171, A010051, A033987, A346482.

Sequence in context: A295663 A005926 A353419 * A349378 A331918 A089803

Adjacent sequences: A346480 A346481 A346482 * A346484 A346485 A346486

Keyword

sign

Author

Mats Granvik and Antti Karttunen, Aug 17 2021

Status

approved