A329445 Dirichlet inverse of A328745.

1, -2, -3, 1, -5, 6, -7, 0, 3, 10, -11, -3, -13, 14, 15, 0, -17, -6, -19, -5, 21, 22, -23, 0, 10, 26, -1, -7, -29, -30, -31, 0, 33, 34, 35, 3, -37, 38, 39, 0, -41, -42, -43, -11, -15, 46, -47, 0, 21, -20, 51, -13, -53, 2, 55, 0, 57, 58, -59, 15, -61, 62, -21, 0, 65, -66, -67, -17

Offset

1,2

Comments

Signed version of A182938.

The asymptotic density of 0's in this sequence is 1 - Product_{p prime} (1 - 1/p^(p+1)) = 0.13585792767780221591... . - Amiram Eldar, Nov 24 2025

Links

Aloe Poliszuk, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = (-1)^e*binomial(p,e) for prime p and e >= 0.

Dirichlet g.f.: Sum_{n>0} a(n)/n^s = Product_{p prime} (1-p^(-s))^p.

a(n) = A182938(n) * A008836(n) for n > 0.

Mathematica

f[p_, e_] := (-1)^e * Binomial[p, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 24 2025 *)

Prog

(Python)
from math import prod, comb
from sympy import factorint
def A329445(n): return prod(-comb(p, e) if e&1 else comb(p, e) for p, e in factorint(n).items()) # Chai Wah Wu, Dec 23 2022
(PARI) a(n) = my(f=factorint(n)); (-1)^bigomega(n)*prod(X=1, #f~, binomial(f[X, 1], f[X, 2])); \\ Aloe Poliszuk, Nov 15 2025

Crossrefs

Cf. A008836, A182938, A328745.

Sequence in context: A130508 A341635 A182938 * A362248 A385212 A055231

Adjacent sequences: A329442 A329443 A329444 * A329446 A329447 A329448

Keyword

sign,mult,easy

Author

Werner Schulte, Nov 13 2019

Status

approved