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A334743 a(1) = 1; a(n) = -Sum_{d|n, d < n} omega(n/d) * a(d), where omega = A001221.

%I A334743 #12 Feb 16 2025 08:34:00
%S A334743 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 A334743 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 A334743 -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 A334743 a(1) = 1; a(n) = -Sum_{d|n, d < n} omega(n/d) * a(d), where omega = A001221.
%C A334743 Dirichlet inverse of A087802. - _Antti Karttunen_, Nov 29 2024
%H A334743 Antti Karttunen, <a href="/A334743/b334743.txt">Table of n, a(n) for n = 1..20000</a>
%H A334743 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>
%F A334743 G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} omega(k) * A(x^k).
%F A334743 Dirichlet g.f.: 1 / (1 + zeta(s) * primezeta(s)).
%t A334743 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 A334743 (PARI)
%o A334743 memoA334743 = Map();
%o A334743 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 A334743 Cf. A001221, A007427, A008480, A008683, A010051, A087802 (Dirichlet inverse), A327276, A334744.
%K A334743 sign
%O A334743 1,30
%A A334743 _Ilya Gutkovskiy_, May 09 2020