A361019 Dirichlet inverse of A038712.

1, -3, -1, 2, -1, 3, -1, 0, 0, 3, -1, -2, -1, 3, 1, 0, -1, 0, -1, -2, 1, 3, -1, 0, 0, 3, 0, -2, -1, -3, -1, 0, 1, 3, 1, 0, -1, 3, 1, 0, -1, -3, -1, -2, 0, 3, -1, 0, 0, 0, 1, -2, -1, 0, 1, 0, 1, 3, -1, 2, -1, 3, 0, 0, 1, -3, -1, -2, 1, -3, -1, 0, -1, 3, 0, -2, 1, -3, -1, 0, 0, 3, -1, 2, 1, 3, 1, 0, -1, 0, 1, -2, 1, 3, 1, 0, -1

Offset

1,2

Comments

Multiplicative because A038712 is.

Links

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

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A038712(n/d) * a(d).

Multiplicative with a(2) = -3, a(2^2) = 2, and a(2^e) = 0 for e > 2, and for odd prime p, a(p)= -1 and a(p^e) = 0 for e > 1. - Amiram Eldar, Mar 02 2023

From Ridouane Oudra, Sep 16 2025: (Start)

a(n) = 2*A008683(n) - A086831(n).

a(n) = 3*A008683(n) - 2*A359591(n).

a(2*n) = A008683(2*n) - 2*A008683(n).

a(2*n+1) = A008683(2*n+1).

The only possible values of a(n) are: 0, +-1, +-2, +-3. More precisely:

a(n) = 0 iff n is in A046790.

a(n) = +-1 iff n is in A056911.

a(n) = +-2 iff n is in A081770.

a(n) = +-3 iff n is in A039956. (End)

Mathematica

f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := If[e < 3, If[e == 1, -3, 2], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 02 2023 *)

Prog

(PARI)
A038712(n) = ((1<<(1+valuation(n, 2)))-1);
memoA361019 = Map();
A361019(n) = if(1==n, 1, my(v); if(mapisdefined(memoA361019, n, &v), v, v = -sumdiv(n, d, if(d<n, A038712(n/d)*A361019(d), 0)); mapput(memoA361019, n, v); (v)));

Crossrefs

Cf. A038712, A008683, A086831, A359591, A046790, A056911, A081770, A039956.

Sequence in context: A352522 A256262 A332929 * A157520 A325116 A227339

Adjacent sequences: A361016 A361017 A361018 * A361020 A361021 A361022

Keyword

sign,mult

Author

Antti Karttunen, Mar 02 2023

Status

approved