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A353336 Sum of A353420 and its Dirichlet inverse.

%I A353336 #12 Apr 20 2022 15:47:14
%S A353336 2,0,0,1,0,4,0,1,4,6,0,2,0,8,12,1,0,14,0,3,16,10,0,2,9,12,28,4,0,12,0,
%T A353336 1,20,14,24,9,0,16,24,3,0,22,0,5,66,20,0,2,16,25,28,6,0,56,30,4,32,22,
%U A353336 0,12,0,26,100,1,36,24,0,7,40,28,0,9,0,28,86,8,40,34,0,3,157,30,0,19,42,32,44,5,0,52,48
%N A353336 Sum of A353420 and its Dirichlet inverse.
%C A353336 The first negative term is a(255255) = -11936.
%H A353336 Antti Karttunen, <a href="/A353336/b353336.txt">Table of n, a(n) for n = 1..16384</a>
%H A353336 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353336 a(n) = A353420(n) + A353335(n).
%F A353336 For n > 1, a(n) = -Sum_{d|n, 1<d<n} A353420(d) * A353335(n/d).
%o A353336 (PARI)
%o A353336 up_to = 65537;
%o A353336 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 A353336 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A353336 A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760
%o A353336 A353420(n) = A126760(A003961(n));
%o A353336 v353335 = DirInverseCorrect(vector(up_to,n,A353420(n)));
%o A353336 A353335(n) = v353335[n];
%o A353336 A353336(n) = (A353420(n)+A353335(n));
%Y A353336 Cf. A003961, A126760, A353420 (also a quadrisection of this sequence), A353335.
%Y A353336 Cf. also A323882, A323894, A349135.
%K A353336 sign
%O A353336 1,1
%A A353336 _Antti Karttunen_, Apr 20 2022