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A353349 Sum of A353350 and its Dirichlet inverse.

%I A353349 #12 Apr 16 2022 09:40:42
%S A353349 2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A353349 0,1,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%U A353349 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2
%N A353349 Sum of A353350 and its Dirichlet inverse.
%C A353349 a(720) = -1 is the first negative term.
%H A353349 Antti Karttunen, <a href="/A353349/b353349.txt">Table of n, a(n) for n = 1..65537</a>
%H A353349 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353349 a(n) = A353348(n) + A353350(n).
%F A353349 a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A353348(d) * A353350(n/d).
%F A353349 a(p) = 0 for all primes p.
%F A353349 a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%t A353349 f[p_, e_] := e*2^(PrimePi[p] - 1); s[1] = 1; s[n_] := Boole@Divisible[Plus @@ f @@@ FactorInteger[n], 3]; sinv[1] = 1; sinv[n_] := -DivisorSum[n, sinv[#]*s[n/#] &, # < n &]; a[n_] := s[n] + sinv[n]; Array[a, 100] (* _Amiram Eldar_, Apr 15 2022 *)
%o A353349 (PARI)
%o A353349 up_to = 16384;
%o A353349 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 A353349 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A353349 A353350(n) = (0==(A048675(n)%3));
%o A353349 v353348 = DirInverseCorrect(vector(up_to,n,A353350(n)));
%o A353349 A353348(n) = v353348[n];
%o A353349 A353349(n) = (A353348(n)+A353350(n));
%Y A353349 Cf. A003961, A048675, A348717, A353348, A353350.
%K A353349 sign
%O A353349 1,1
%A A353349 _Antti Karttunen_, Apr 15 2022