A359551 Dirichlet inverse of A359550, which is multiplicative sequence with a(p^e) = 1 if e < p, otherwise 0.

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

Offset

1

Comments

Multiplicative because A359550 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} A359550(n/d) * a(d).

Multiplicative with a(p^e) = -1 if e == 1 (mod p), 1 if e == 0 (mod p), and 0 otherwise. - Amiram Eldar, Jan 06 2023

Mathematica

f[p_, e_] := If[(r = Mod[e, p]) == 1, -1, If[r == 0, 1, 0]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 06 2023 *)

Prog

(PARI) A359551(n) = { my(f=factor(n), r); prod(i=1, #f~, r = f[i, 2]%f[i, 1]; if(r>1, 0, (-1)^r)); }; \\ After Amiram Eldar's multiplicative formula and Mathematica-code.
(Python)
from math import prod
from sympy import factorint
def A359551(n): return prod(1 if (d:=e%p)==0 else (-1 if d==1 else 0) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 06 2023

Crossrefs

Cf. A359550.

Cf. also A359432.

Sequence in context: A355689 A353627 A353628 * A353458 A110242 A356315

Adjacent sequences: A359548 A359549 A359550 * A359552 A359553 A359554

Keyword

sign,mult

Author

Antti Karttunen, Jan 06 2023

Status

approved