A349914 Sum of A000593 (the sum of odd divisors function) and its Dirichlet inverse.

2, 0, 0, 1, 0, 8, 0, 1, 16, 12, 0, 4, 0, 16, 48, 1, 0, 10, 0, 6, 64, 24, 0, 4, 36, 28, 40, 8, 0, 0, 0, 1, 96, 36, 96, 13, 0, 40, 112, 6, 0, 0, 0, 12, 60, 48, 0, 4, 64, 26, 144, 14, 0, 40, 144, 8, 160, 60, 0, 24, 0, 64, 80, 1, 168, 0, 0, 18, 192, 0, 0, 13, 0, 76, 104, 20, 192, 0, 0, 6, 121, 84, 0, 32, 216, 88, 240

Offset: 1

Author

Antti Karttunen, Dec 08 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A000593 (also a quadrisection of this sequence), A327278.

Cf. also A349913, A349916.

Programs

Mathematica

f1[p_,e_] := If[p==2, 1, (p^(e+1)-1)/(p-1)]; f2[p_, e_] := If[p == 2, -Boole[e == 1], Which[e == 1, -p - 1, e == 2, p, 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

A000593(n) = sigma(n>>valuation(n, 2));
A327278(n) = sumdiv(n,d,if(d%2,d*moebius(d)*moebius(n/d),0));
A349914(n) = (A000593(n)+A327278(n));

Formula

a(n) = A000593(n) + A327278(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A000593(d) * A327278(n/d).

a(4*n) = A000593(n).