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A349445 Dirichlet convolution of A001511 (the 2-adic valuation of 2n) with A349134 (the Dirichlet inverse of Kimberling's paraphrases).

%I A349445 #12 Nov 21 2021 01:19:45
%S A349445 1,1,-1,1,-2,-1,-3,1,-2,-2,-5,-1,-6,-3,0,1,-8,-2,-9,-2,0,-5,-11,-1,-6,
%T A349445 -6,-4,-3,-14,0,-15,1,0,-8,0,-2,-18,-9,0,-2,-20,0,-21,-5,2,-11,-23,-1,
%U A349445 -12,-6,0,-6,-26,-4,0,-3,0,-14,-29,0,-30,-15,3,1,0,0,-33,-8,0,0,-35,-2,-36,-18,4,-9,0,0,-39,-2,-8
%N A349445 Dirichlet convolution of A001511 (the 2-adic valuation of 2n) with A349134 (the Dirichlet inverse of Kimberling's paraphrases).
%H A349445 Antti Karttunen, <a href="/A349445/b349445.txt">Table of n, a(n) for n = 1..20000</a>
%F A349445 a(n) = Sum_{d|n} A001511(n/d) * A349134(d).
%F A349445 If p odd prime, a(p) = (1-p)/2. - _Bernard Schott_, Nov 19 2021
%t A349445 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, IntegerExponent[2*#, 2]*kinv[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2021 *)
%o A349445 (PARI)
%o A349445 A001511(n) = (1+valuation(n,2));
%o A349445 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349445 memoA349134 = Map();
%o A349445 A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(d<n,A003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
%o A349445 A349445(n) = sumdiv(n,d,A001511(n/d)*A349134(d));
%Y A349445 Cf. A001511, A003602, A349134, A349444 (Dirichlet inverse), A349446 (sum with it).
%Y A349445 Cf. also A349432, A349448.
%K A349445 sign
%O A349445 1,5
%A A349445 _Antti Karttunen_, Nov 18 2021