A359378 Dirichlet inverse of A359377, where A359377(n) = 1 if 3*n is squarefree, otherwise 0.

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

Offset

1

Comments

Note the correspondences between four sequences:

A156277 --- abs ---> A359377

^ ^

| |

inv inv

| |

v v

A011655 <--- abs --- A359378 (this sequence)

Here inv means that the sequences are Dirichlet Inverses of each other, and abs means taking absolute values.

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} A359377(n/d) * a(d).

Multiplicative with a(3^e) = 0 and a(p^k) = (-1)^k for all primes p <> 3.

a(n) = A359170(n) - A359172(n).

For all n >= 1, a(A001651(n)) = A008836(A001651(n)).

Dirichlet g.f.: 3^s/((3^s-1)*zeta(s)). - Amiram Eldar, Jan 03 2023

Mathematica

f[p_, e_] := (-1)^e; f[3, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 30 2022 *)

Prog

(PARI) A359378(n) = { my(f = factor(n)); prod(k=1, #f~, (3!=f[k, 1])*((-1)^f[k, 2])); };

Crossrefs

Cf. A001651, A008836, A011655 (absolute values), A156277 (Dirichlet inverse of the absolute values), A359377 (Dirichlet inverse).

Cf. A008585 (after its initial term gives the positions of 0's), A359171 (of positive terms), A359381 (of negative terms), A359170, A359172.

Cf. also A166698, A358839.

Sequence in context: A049347 A010892 A091338 * A016345 A016148 A016333

Adjacent sequences: A359375 A359376 A359377 * A359379 A359380 A359381

Keyword

sign,mult

Author

Antti Karttunen, Dec 29 2022

Status

approved