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A349353 Dirichlet inverse of A328203.

%I A349353 #9 Nov 21 2021 10:16:42
%S A349353 1,-2,-5,0,-8,10,-11,0,5,16,-17,0,-20,22,38,0,-26,-10,-29,0,52,34,-35,
%T A349353 0,11,40,1,0,-44,-76,-47,0,80,52,82,0,-56,58,94,0,-62,-104,-65,0,-34,
%U A349353 70,-71,0,19,-22,122,0,-80,-2,126,0,136,88,-89,0,-92,94,-46,0,148,-160,-101,0,164,-164,-107,0,-110,112,-45
%N A349353 Dirichlet inverse of A328203.
%H A349353 Antti Karttunen, <a href="/A349353/b349353.txt">Table of n, a(n) for n = 1..20000</a>
%F A349353 a(1) = 1; a(n) = -Sum_{d|n, d < n} A328203(n/d) * a(d).
%F A349353 a(n) = A349354(n) - A328203(n).
%F A349353 a(n) = Sum_{d|n} A349134(d) * A349341(n/d).
%F A349353 a(n) = Sum_{d|n} A349343(d) * A349344(n/d).
%o A349353 (PARI)
%o A349353 up_to = 20000;
%o A349353 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 A349353 A328203(n) = if(n%2,(1/2)*(sigma(n)+(n*numdiv(n))),2*A328203(n/2));
%o A349353 v349353 = DirInverseCorrect(vector(up_to,n,A328203(n)));
%o A349353 A349353(n) = v349353[n];
%Y A349353 Dirichlet convolution of A349134 with A349341, or equally of A349343 with A349344.
%Y A349353 Cf. A328203, A349354.
%K A349353 sign
%O A349353 1,2
%A A349353 _Antti Karttunen_, Nov 15 2021