A365428 Dirichlet inverse of A102283.

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

Offset

1

Comments

Multiplicative because A102283 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} A102283(n/d) * a(d).

Multiplicative with a(p) = -Legendre(-3, p), and a(p^e) = 0 for e >= 2. - Amiram Eldar, Sep 16 2023

Mathematica

f[p_, e_] := If[e == 1, -JacobiSymbol[-3, p], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2023 *)

Prog

(PARI) A365428(n) = { my(f=factor(n)); prod(k=1, #f~, if(1<f[k, 2], 0, -kronecker(-3, f[k, 1]))); }; \\ (After Amiram Eldar's multiplicative formula).
(PARI)
A102283(n) = ([0, 1, -1][n%3 + 1]);
memoA365428 = Map();
A365428(n) = if(1==n, 1, my(v); if(mapisdefined(memoA365428, n, &v), v, v = -sumdiv(n, d, if(d<n, A102283(n/d)*A365428(d), 0)); mapput(memoA365428, n, v); (v)));

Crossrefs

Cf. A102283, A134323, A156277, A359377 (absolute values).

Sequence in context: A074711 A004585 A319448 * A156277 A359377 A353663

Adjacent sequences: A365425 A365426 A365427 * A365429 A365430 A365431

Keyword

sign,mult

Author

Antti Karttunen, Sep 16 2023

Status

approved