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A353418 Dirichlet inverse of A353269.

%I A353418 #9 Apr 19 2022 22:45:30
%S A353418 1,0,0,-1,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,-1,0,0,0,0,-1,0,
%T A353418 0,0,-1,0,1,0,0,-1,1,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,-1,0,-1,0,0,0,0,-1,
%U A353418 0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,-1,0,-1,1,0,1,-1,0,0,-1,0,0,0,0,0,2
%N A353418 Dirichlet inverse of A353269.
%H A353418 Antti Karttunen, <a href="/A353418/b353418.txt">Table of n, a(n) for n = 1..65537</a>
%H A353418 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353418 a(1) = 1; a(n) = -Sum_{d|n, d < n} A353269(n/d) * a(d).
%F A353418 a(n) = A353419(n) - A353269(n).
%F A353418 a(p) = 0 for all primes p.
%F A353418 a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%o A353418 (PARI)
%o A353418 up_to = 65537;
%o A353418 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v
%o A353418 A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
%o A353418 A353269(n) = (!(A156552(n)%3));
%o A353418 v353418 = DirInverseCorrect(vector(up_to,n,A353269(n)));
%o A353418 A353418(n) = v353418[n];
%Y A353418 Cf. A003961, A156552, A348717, A353269, A353419.
%Y A353418 Cf. also A353348, A353422.
%K A353418 sign
%O A353418 1,100
%A A353418 _Antti Karttunen_, Apr 19 2022