%I #12 Nov 21 2021 01:19:45

%S 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 -6,-4,-3,-14,0,-15,1,0,-8,0,-2,-18,-9,0,-2,-20,0,-21,-5,2,-11,-23,-1,

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

%H Antti Karttunen, <a href="/A349445/b349445.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = Sum_{d|n} A001511(n/d) * A349134(d).

%F If p odd prime, a(p) = (1-p)/2. - _Bernard Schott_, Nov 19 2021

%t 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 (PARI)

%o A001511(n) = (1+valuation(n,2));

%o A003602(n) = (1+(n>>valuation(n,2)))/2;

%o memoA349134 = Map();

%o 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(n) = sumdiv(n,d,A001511(n/d)*A349134(d));

%Y Cf. A001511, A003602, A349134, A349444 (Dirichlet inverse), A349446 (sum with it).

%Y Cf. also A349432, A349448.

%K sign

%O 1,5

%A _Antti Karttunen_, Nov 18 2021