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

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, 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, -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

Offset: 1

Author

Antti Karttunen, Nov 19 2021

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

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

Cf. also A349432, A349445.

Programs

Mathematica

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 *)

PARI

A000265(n) = (n >> valuation(n, 2));
A003602(n) = (1+(n>>valuation(n,2)))/2;
memoA349134 = Map();
A349134(n) = if(1==n,1,my(v); if(mapisdefined(memoA349134,n,&v), v, v = -sumdiv(n,d,if(dA003602(n/d)*A349134(d),0)); mapput(memoA349134,n,v); (v)));
A349448(n) = sumdiv(n,d,A000265(d)*A349134(n/d));

Formula

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

From Bernard Schott, Dec 18 2021: (Start)

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

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