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A349448 Dirichlet convolution of A000265 (odd part of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).

%I A349448 #17 Dec 18 2021 23:38:42
%S A349448 1,0,1,0,2,0,3,0,2,0,5,0,6,0,0,0,8,0,9,0,0,0,11,0,6,0,4,0,14,0,15,0,0,
%T A349448 0,0,0,18,0,0,0,20,0,21,0,-2,0,23,0,12,0,0,0,26,0,0,0,0,0,29,0,30,0,
%U A349448 -3,0,0,0,33,0,0,0,35,0,36,0,-4,0,0,0,39,0,8,0,41,0,0,0,0,0,44,0,0,0,0,0,0,0,48,0
%N A349448 Dirichlet convolution of A000265 (odd part of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).
%H A349448 Antti Karttunen, <a href="/A349448/b349448.txt">Table of n, a(n) for n = 1..20000</a>
%F A349448 a(n) = Sum_{d|n} A000265(d) * A349134(n/d).
%F A349448 From _Bernard Schott_, Dec 18 2021: (Start)
%F A349448 If p is an odd prime, a(p) = (p-1)/2.
%F A349448 If n is even, a(n) = 0. (End)
%t A349448 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, # / 2^IntegerExponent[#, 2] * kinv[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2021 *)
%o A349448 (PARI)
%o A349448 A000265(n) = (n >> valuation(n, 2));
%o A349448 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349448 memoA349134 = Map();
%o A349448 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 A349448 A349448(n) = sumdiv(n,d,A000265(d)*A349134(n/d));
%Y A349448 Cf. A000265, A003602, A349134, A349447 (Dirichlet inverse).
%Y A349448 Cf. also A349432, A349445.
%K A349448 sign
%O A349448 1,5
%A A349448 _Antti Karttunen_, Nov 19 2021