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A353419 Sum of A353269 and its Dirichlet inverse.

%I A353419 #8 Apr 19 2022 22:45:56
%S A353419 2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A353419 0,2,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%U A353419 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,2,0,2,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2
%N A353419 Sum of A353269 and its Dirichlet inverse.
%C A353419 The first negative term is a(840) = -2.
%H A353419 Antti Karttunen, <a href="/A353419/b353419.txt">Table of n, a(n) for n = 1..65537</a>
%H A353419 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353419 a(n) = A353269(n) + A353418(n).
%F A353419 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A353269(d) * A353418(n/d).
%F A353419 a(p) = 0 for all primes p.
%F A353419 a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%o A353419 (PARI)
%o A353419 up_to = 65537;
%o A353419 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 A353419 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 A353419 A353269(n) = (!(A156552(n)%3));
%o A353419 v353418 = DirInverseCorrect(vector(up_to,n,A353269(n)));
%o A353419 A353418(n) = v353418[n];
%o A353419 A353419(n) = (A353269(n)+A353418(n));
%Y A353419 Cf. A003961, A156552, A348717, A353269, A353418.
%Y A353419 Cf. also A353349.
%K A353419 sign
%O A353419 1,1
%A A353419 _Antti Karttunen_, Apr 19 2022