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

1,1

Links

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

Formula

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

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

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

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 *)

Prog

(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));

Crossrefs

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

Cf. also A349913, A349916.

Sequence in context: A349916 A349342 A365712 * A354105 A370366 A342419

Adjacent sequences: A349911 A349912 A349913 * A349915 A349916 A349917

Keyword

nonn

Author

Antti Karttunen, Dec 08 2021

Status

approved