A334744 a(1) = 1; a(n) = -Sum_{d|n, d < n} bigomega(n/d) * a(d), where bigomega = A001222.

1, -1, -1, -1, -1, 0, -1, 0, -1, 0, -1, 2, -1, 0, 0, 1, -1, 2, -1, 2, 0, 0, -1, 2, -1, 0, 0, 2, -1, 3, -1, 1, 0, 0, 0, 2, -1, 0, 0, 2, -1, 3, -1, 2, 2, 0, -1, -1, -1, 2, 0, 2, -1, 2, 0, 2, 0, 0, -1, 0, -1, 0, 2, 0, 0, 3, -1, 2, 0, 3, -1, -3, -1, 0, 2, 2, 0, 3, -1, -1, 1, 0, -1, 0, 0, 0, 0, 2, -1, 0, 0, 2

Offset: 1

Author

Ilya Gutkovskiy, May 09 2020

Keywords

• sign

Comments

Dirichlet inverse of A086436. - Antti Karttunen, Nov 29 2024

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Eric Weisstein's World of Mathematics, Prime Factor

Crossrefs

Cf. A001222, A007427, A069513, A086436 (Dirichlet inverse), A327276, A334743.

Programs

Mathematica

a[n_] := If[n == 1, n, -Sum[If[d < n, PrimeOmega[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 92}]

PARI

memoA334744 = Map();
A334744(n) = if(1==n,1,my(v); if(mapisdefined(memoA334744,n,&v), v, v = -sumdiv(n,d,if(dA334744(d),0)); mapput(memoA334744,n,v); (v))); \\ Antti Karttunen, Nov 29 2024

Formula

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} bigomega(k) * A(x^k).

Dirichlet g.f.: 1 / (1 + zeta(s) * Sum_{k>=1} primezeta(k*s)).