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

%I A334744 #11 Feb 16 2025 08:34:00
%S A334744 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,
%T A334744 -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,
%U A334744 -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
%N A334744 a(1) = 1; a(n) = -Sum_{d|n, d < n} bigomega(n/d) * a(d), where bigomega = A001222.
%C A334744 Dirichlet inverse of A086436. - _Antti Karttunen_, Nov 29 2024
%H A334744 Antti Karttunen, <a href="/A334744/b334744.txt">Table of n, a(n) for n = 1..20000</a>
%H A334744 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>
%F A334744 G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} bigomega(k) * A(x^k).
%F A334744 Dirichlet g.f.: 1 / (1 + zeta(s) * Sum_{k>=1} primezeta(k*s)).
%t A334744 a[n_] := If[n == 1, n, -Sum[If[d < n, PrimeOmega[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 92}]
%o A334744 (PARI)
%o A334744 memoA334744 = Map();
%o A334744 A334744(n) = if(1==n,1,my(v); if(mapisdefined(memoA334744,n,&v), v, v = -sumdiv(n,d,if(d<n,bigomega(n/d)*A334744(d),0)); mapput(memoA334744,n,v); (v))); \\ _Antti Karttunen_, Nov 29 2024
%Y A334744 Cf. A001222, A007427, A069513, A086436 (Dirichlet inverse), A327276, A334743.
%K A334744 sign
%O A334744 1,12
%A A334744 _Ilya Gutkovskiy_, May 09 2020