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A349435 Dirichlet convolution of A230593 with A347084, which is Dirichlet inverse of {n + its arithmetic derivative}.

%I A349435 #13 Nov 21 2021 01:19:00
%S A349435 1,0,0,-2,0,0,0,-2,-3,0,0,2,0,0,0,-2,0,3,0,2,0,0,0,0,-5,0,-6,2,0,0,0,
%T A349435 -2,0,0,0,7,0,0,0,0,0,0,0,2,3,0,0,-2,-7,5,0,2,0,3,0,0,0,0,0,-4,0,0,3,
%U A349435 -2,0,0,0,2,0,0,0,5,0,0,5,2,0,0,0,-2,-12,0,0,-4,0,0,0,0,0,-6,0,2,0,0,0,-4,0,7,3
%N A349435 Dirichlet convolution of A230593 with A347084, which is Dirichlet inverse of {n + its arithmetic derivative}.
%C A349435 Dirichlet convolution of this sequence with A348976 is A349338.
%C A349435 The positions of records start as: 1, 12, 18, 36, 100, 108, 196, 225, 324, 441, 500, 1125, 1372, 2500, 5000, 5324, 8575, 8788, 9604, 12500, 19652, etc.
%H A349435 Antti Karttunen, <a href="/A349435/b349435.txt">Table of n, a(n) for n = 1..20000</a>
%F A349435 a(n) = Sum_{d|n} A230593(n/d) * A347084(d).
%t A349435 s[n_] := n * DivisorSum[n, 1/# &, !CompositeQ[#] &]; f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); dinv[1] = 1; dinv[n_] := dinv[n] = -DivisorSum[n, dinv[#] * d[n/#] &, # < n &]; a[n_] := DivisorSum[n, s[#] * dinv[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 18 2021 *)
%o A349435 (PARI)
%o A349435 up_to = 20000;
%o A349435 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 A349435 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A349435 A230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
%o A349435 v347084 = DirInverseCorrect(vector(up_to,n,n+A003415(n)));
%o A349435 A347084(n) = v347084[n];
%o A349435 A349435(n) = sumdiv(n,d,A230593(n/d)*A347084(d));
%Y A349435 Cf. A003415, A129283, A230593, A347084, A349434 (Dirichlet inverse), A349436 (sum with it).
%Y A349435 Cf. also A348976, A349338.
%K A349435 sign
%O A349435 1,4
%A A349435 _Antti Karttunen_, Nov 17 2021