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A353421 Dirichlet convolution of A353269 with A353348 (the Dirichlet inverse of A353350).

%I A353421 #9 Apr 19 2022 22:45:47
%S A353421 1,0,0,1,0,-1,0,-1,1,1,0,0,0,-1,-1,1,0,0,0,-1,1,1,0,-1,1,-1,-1,0,0,1,
%T A353421 0,-1,-1,1,-1,1,0,-1,1,1,0,0,0,-1,0,1,0,1,1,-1,-1,0,0,-1,1,-1,1,-1,0,
%U A353421 -2,0,1,-1,1,-1,1,0,-1,-1,0,0,-1,0,-1,0,0,-1,0,0,-2,1,1,0,2,1,-1,1,1,0,2,1,-1,-1,1,-1
%N A353421 Dirichlet convolution of A353269 with A353348 (the Dirichlet inverse of A353350).
%C A353421 Dirichlet convolution between this sequence and A353352 is A353362.
%H A353421 Antti Karttunen, <a href="/A353421/b353421.txt">Table of n, a(n) for n = 1..65537</a>
%H A353421 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353421 a(n) = Sum_{d|n} A353269(n/d) * A353348(d).
%F A353421 a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%o A353421 (PARI)
%o A353421 up_to = 65537;
%o A353421 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 (correctly!)
%o A353421 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A353421 A353350(n) = (0==(A048675(n)%3));
%o A353421 v353348 = DirInverseCorrect(vector(up_to,n,A353350(n)));
%o A353421 A353348(n) = v353348[n];
%o A353421 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 A353421 A353269(n) = (!(A156552(n)%3));
%o A353421 A353421(n) = sumdiv(n,d,A353269(n/d)*A353348(d));
%Y A353421 Cf. A003961, A048675, A156552, A348717, A353269, A353350, A353348.
%Y A353421 Cf. A353422 (Dirichlet inverse).
%Y A353421 Cf. A353352, A353362.
%K A353421 sign
%O A353421 1,60
%A A353421 _Antti Karttunen_, Apr 19 2022