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A349434 Dirichlet convolution of A129283 (n + its arithmetic derivative) with A349337 (Dirichlet inverse of A230593).

%I A349434 #9 Nov 21 2021 01:18:48
%S A349434 1,0,0,2,0,0,0,2,3,0,0,-2,0,0,0,6,0,-3,0,-2,0,0,0,0,5,0,6,-2,0,0,0,10,
%T A349434 0,0,0,5,0,0,0,0,0,0,0,-2,-3,0,0,-6,7,-5,0,-2,0,-3,0,0,0,0,0,4,0,0,-3,
%U A349434 22,0,0,0,-2,0,0,0,-5,0,0,-5,-2,0,0,0,-6,21,0,0,4,0,0,0,0,0,6,0,-2,0,0,0,-4,0,-7,-3,7
%N A349434 Dirichlet convolution of A129283 (n + its arithmetic derivative) with A349337 (Dirichlet inverse of A230593).
%C A349434 Dirichlet convolution of this sequence with A349338 is A348976.
%H A349434 Antti Karttunen, <a href="/A349434/b349434.txt">Table of n, a(n) for n = 1..20000</a>
%F A349434 a(n) = Sum_{d|n} A129283(n/d) * A349337(d).
%t A349434 s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); a[n_] := DivisorSum[n, sinv[#] * d[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 18 2021 *)
%o A349434 (PARI)
%o A349434 up_to = 20000;
%o A349434 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 A349434 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A349434 A129283(n) = (n+A003415(n));
%o A349434 A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
%o A349434 v349337 = DirInverseCorrect(vector(up_to,n,A230593(n)));
%o A349434 A349337(n) = v349337[n];
%o A349434 A349434(n) = sumdiv(n,d,A129283(n/d)*A349337(d));
%Y A349434 Cf. A003415, A129283, A230593, A349337, A349435 (Dirichlet inverse), A349436 (sum with it).
%Y A349434 Cf. also A348976, A349338.
%K A349434 sign
%O A349434 1,4
%A A349434 _Antti Karttunen_, Nov 17 2021