A349913 Sum of A001227 (the number of odd divisors function) and its Dirichlet inverse.

2, 0, 0, 1, 0, 4, 0, 1, 4, 4, 0, 2, 0, 4, 8, 1, 0, 2, 0, 2, 8, 4, 0, 2, 4, 4, 4, 2, 0, 0, 0, 1, 8, 4, 8, 3, 0, 4, 8, 2, 0, 0, 0, 2, 4, 4, 0, 2, 4, 2, 8, 2, 0, 4, 8, 2, 8, 4, 0, 4, 0, 4, 4, 1, 8, 0, 0, 2, 8, 0, 0, 3, 0, 4, 4, 2, 8, 0, 0, 2, 5, 4, 0, 4, 8, 4, 8, 2, 0, 8, 8, 2, 8, 4, 8, 2, 0, 2, 4, 3, 0, 0, 0, 2, 0

Offset: 1

Author

Antti Karttunen, Dec 08 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A001227 (also a quadrisection of this sequence), A327276.

Cf. also A349914, A349916.

Programs

Mathematica

f1[p_,e_] := If[p==2, 1, e+1]; f2[p_, e_] := Which[e == 1, -1 - Boole[p > 2], e == 2, Boole[p > 2], e > 2, 0]; a[1] = 2; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) + Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Dec 08 2021 *)

PARI

A001227(n) = numdiv(n>>valuation(n, 2));
A327276(n) = sumdiv(n, d, if(d%2, moebius(d)*moebius(n/d))); \\ From A327276
A349913(n) = (A001227(n)+A327276(n));

Formula

a(n) = A001227(n) + A327276(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A001227(d) * A327276(n/d).

a(4*n) = A001227(n).