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A349432 Dirichlet convolution of A000027 (the identity function) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

%I A349432 #10 Nov 21 2021 01:18:31
%S A349432 1,1,1,2,2,1,3,4,2,2,5,2,6,3,0,8,8,2,9,4,0,5,11,4,6,6,4,6,14,0,15,16,
%T A349432 0,8,0,4,18,9,0,8,20,0,21,10,-2,11,23,8,12,6,0,12,26,4,0,12,0,14,29,0,
%U A349432 30,15,-3,32,0,0,33,16,0,0,35,8,36,18,-4,18,0,0,39,16,8,20,41,0,0,21,0,20,44,-2,0,22,0,23
%N A349432 Dirichlet convolution of A000027 (the identity function) with A349134 (Dirichlet inverse of Kimberling's paraphrases).
%H A349432 Antti Karttunen, <a href="/A349432/b349432.txt">Table of n, a(n) for n = 1..20000</a>
%t A349432 k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; kinv[1] = 1; kinv[n_] := kinv[n] = -DivisorSum[n, kinv[#] * k[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * kinv[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 18 2021 *)
%o A349432 (PARI)
%o A349432 up_to = 16384;
%o A349432 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 A349432 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349432 v349134 = DirInverseCorrect(vector(up_to,n,A003602(n)));
%o A349432 A349134(n) = v349134[n];
%o A349432 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349432 A055615(n) = (n*moebius(n));
%o A349432 A349432(n) = sumdiv(n,d,d*A349134(n/d));
%Y A349432 Cf. A003602, A055615, A349134, A349431 (Dirichlet inverse), A349433 (sum with it).
%Y A349432 Cf. also A349445, A349448.
%K A349432 sign
%O A349432 1,4
%A A349432 _Antti Karttunen_, Nov 17 2021