A351619 a(n) = Sum_{p|n, p prime} (-1)^p.

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

Offset: 1

Author

Seiichi Manyama, Mar 02 2022

Keywords

• sign

Links

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

Crossrefs

Cf. A001221, A048272, A305614.

Programs

Mathematica

A351619[n_] := 2*Boole[EvenQ[n]] - PrimeNu[n]; Array[A351619, 100] (* Paolo Xausa, Jan 28 2025 *)

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, (-1)^f[k, 1]);

PARI

my(N=99, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, isprime(k)*(-x)^k/(1-x^k))))

Python

from sympy import primefactors
def A351619(n): return (0 if n%2 else 2) - len(primefactors(n)) # Chai Wah Wu, Mar 02 2022

Formula

G.f.: Sum_{k>=1} (-x)^prime(k)/(1 - x^prime(k)).

a(n) = -A001221(n) if n is odd and a(n) = 2 - A001221(n) if n is even. - Chai Wah Wu, Mar 02 2022