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

1,5

Links

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

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)

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 *)

Prog

(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(d<n, A003602(n/d)*A349134(d), 0)); mapput(memoA349134, n, v); (v)));
A349448(n) = sumdiv(n, d, A000265(d)*A349134(n/d));

Crossrefs

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

Cf. also A349432, A349445.

Sequence in context: A128143 A292561 A027640 * A194666 A325799 A355930

Adjacent sequences: A349445 A349446 A349447 * A349449 A349450 A349451

Keyword

sign

Author

Antti Karttunen, Nov 19 2021

Status

approved