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A349444 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A092673 (Dirichlet inverse of A001511).

%I A349444 #13 Dec 14 2021 12:13:42
%S A349444 1,-1,1,0,2,-1,3,0,3,-2,5,0,6,-3,4,0,8,-3,9,0,6,-5,11,0,10,-6,9,0,14,
%T A349444 -4,15,0,10,-8,12,0,18,-9,12,0,20,-6,21,0,12,-11,23,0,21,-10,16,0,26,
%U A349444 -9,20,0,18,-14,29,0,30,-15,18,0,24,-10,33,0,22,-12,35,0,36,-18,20,0,30,-12,39,0,27,-20,41,0,32
%N A349444 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A092673 (Dirichlet inverse of A001511).
%H A349444 Antti Karttunen, <a href="/A349444/b349444.txt">Table of n, a(n) for n = 1..20000</a>
%F A349444 a(n) = Sum_{d|n} A003602(n/d) * A092673(d).
%t A349444 s[n_] := MoebiusMu[n] - If[OddQ[n], 0, MoebiusMu[n/2]]; k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, s[#]*k[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2021 *)
%o A349444 (PARI)
%o A349444 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349444 A092673(n) = if(n<1, 0, moebius(n) - if( n%2, 0, moebius(n/2))); \\ From A092673
%o A349444 A349444(n) = sumdiv(n,d,A003602(n/d)*A092673(d));
%Y A349444 Cf. A001511, A003602, A008683, A092673, A349445 (Dirichlet inverse), A349446 (sum with it).
%Y A349444 Cf. also A349431, A349447.
%K A349444 sign
%O A349444 1,5
%A A349444 _Antti Karttunen_, Nov 18 2021