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A349447 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A326937 (Dirichlet inverse of A000265).

%I A349447 #13 Nov 21 2021 01:20:01
%S A349447 1,0,-1,0,-2,0,-3,0,-1,0,-5,0,-6,0,4,0,-8,0,-9,0,6,0,-11,0,-2,0,-1,0,
%T A349447 -14,0,-15,0,10,0,12,0,-18,0,12,0,-20,0,-21,0,4,0,-23,0,-3,0,16,0,-26,
%U A349447 0,20,0,18,0,-29,0,-30,0,6,0,24,0,-33,0,22,0,-35,0,-36,0,4,0,30,0,-39,0,-1,0,-41,0,32,0,28
%N A349447 Dirichlet convolution of A003602 (Kimberling's paraphrases) with A326937 (Dirichlet inverse of A000265).
%C A349447 Dirichlet convolution of this sequence with A264740 is A349371.
%H A349447 Antti Karttunen, <a href="/A349447/b349447.txt">Table of n, a(n) for n = 1..20000</a>
%F A349447 a(n) = Sum_{d|n} A003602(d) * A326937(n/d).
%t A349447 k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, MoebiusMu[#] * # / 2^IntegerExponent[#, 2] * k[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2021 *)
%o A349447 (PARI)
%o A349447 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349447 A006519(n) = (1<<valuation(n, 2));
%o A349447 A055615(n) = (n*moebius(n));
%o A349447 A326937(n) = (A055615(n)/A006519(n));
%o A349447 A349447(n) = sumdiv(n,d,A003602(d)*A326937(n/d));
%Y A349447 Cf. A000265, A003602, A326937, A349448 (Dirichlet inverse).
%Y A349447 Cf. also A264740, A349371, A349431, A349444.
%K A349447 sign
%O A349447 1,5
%A A349447 _Antti Karttunen_, Nov 19 2021