A355688 Dirichlet inverse of A354354, characteristic function of numbers that are neither multiples of 2 nor 3.

1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0

Offset

1

Comments

Multiplicative because A354354 is.

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A354354(n/d) * a(d).

a(n) = A008683(n) * A354354(n). [As the latter is fully multiplicative] - Antti Karttunen, Dec 20 2022

From Amiram Eldar, Dec 27 2022: (Start)

Multiplicative with a(p^e) = 0 if p <= 3 or e >= 2, and -1 otherwise.

Dirichlet g.f.: (2^s)*(3^s)/(zeta(s)*(2^s-1)*(3^s-1)). (End)

Mathematica

a[n_] := If[GCD[n, 6] == 1, MoebiusMu[n], 0]; Array[a, 100] (* Amiram Eldar, Dec 27 2022 *)

Prog

(PARI)
A354354(n) = ((n%2)&&(n%3));
memoA355688 = Map();
A355688(n) = if(1==n, 1, my(v); if(mapisdefined(memoA355688, n, &v), v, v = -sumdiv(n, d, if(d<n, A354354(n/d)*A355688(d), 0)); mapput(memoA355688, n, v); (v)));
(PARI) A355688(n) = (moebius(n)*A354354(n)); \\ Antti Karttunen, Dec 20 2022

Crossrefs

Cf. A007310, A008683, A354354.

Cf. also A355689, A355690.

Sequence in context: A104108 A267056 A089024 * A353488 A232991 A168553

Adjacent sequences: A355685 A355686 A355687 * A355689 A355690 A355691

Keyword

sign,mult

Author

Antti Karttunen, Jul 15 2022

Status

approved