A334743 - OEIS

%I #11 Nov 29 2024 16:33:14

%S 1,-1,-1,0,-1,0,-1,0,0,0,-1,1,-1,0,0,0,-1,1,-1,1,0,0,-1,0,0,0,0,1,-1,

%T 3,-1,0,0,0,0,0,-1,0,0,0,-1,3,-1,1,1,0,-1,0,0,1,0,1,-1,0,0,0,0,0,-1,

%U -1,-1,0,1,0,0,3,-1,1,0,3,-1,-1,-1,0,1,1,0,3,-1,0,0,0,-1,-1,0,0,0,0,-1,-1

%N a(1) = 1; a(n) = -Sum_{d|n, d < n} omega(n/d) * a(d), where omega = A001221.

%C Dirichlet inverse of A087802. - _Antti Karttunen_, Nov 29 2024

%H Antti Karttunen, <a href="/A334743/b334743.txt">Table of n, a(n) for n = 1..20000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>

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

%F Dirichlet g.f.: 1 / (1 + zeta(s) * primezeta(s)).

%t a[n_] := If[n == 1, n, -Sum[If[d < n, PrimeNu[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 90}]

%o (PARI)

%o memoA334743 = Map();

%o A334743(n) = if(1==n,1,my(v); if(mapisdefined(memoA334743,n,&v), v, v = -sumdiv(n,d,if(d<n,omega(n/d)*A334743(d),0)); mapput(memoA334743,n,v); (v))); \\ _Antti Karttunen_, Nov 29 2024

%Y Cf. A001221, A007427, A008480, A008683, A010051, A087802 (Dirichlet inverse), A327276, A334744.

%K sign

%O 1,30

%A _Ilya Gutkovskiy_, May 09 2020