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

1,1

Links

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

Formula

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

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

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

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 *)

Prog

(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));

Crossrefs

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

Cf. also A349914, A349916.

Sequence in context: A054876 A109502 A323887 * A346236 A323365 A349135

Adjacent sequences: A349910 A349911 A349912 * A349914 A349915 A349916

Keyword

nonn

Author

Antti Karttunen, Dec 08 2021

Status

approved