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Dirichlet convolution of A000265 (odd part of n) with A349134 (Dirichlet inverse of Kimberling's paraphrases).
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%I #17 Dec 18 2021 23:38:42

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

%H Antti Karttunen, <a href="/A349448/b349448.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = Sum_{d|n} A000265(d) * A349134(n/d).

%F From _Bernard Schott_, Dec 18 2021: (Start)

%F If p is an odd prime, a(p) = (p-1)/2.

%F If n is even, a(n) = 0. (End)

%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, # / 2^IntegerExponent[#, 2] * kinv[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 19 2021 *)

%o (PARI)

%o A000265(n) = (n >> 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 A349448(n) = sumdiv(n,d,A000265(d)*A349134(n/d));

%Y Cf. A000265, A003602, A349134, A349447 (Dirichlet inverse).

%Y Cf. also A349432, A349445.

%K sign

%O 1,5

%A _Antti Karttunen_, Nov 19 2021