A345221 Number of divisors of n with an even sum of divisors.

0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 3, 1, 2, 3, 0, 1, 2, 1, 3, 3, 2, 1, 4, 1, 2, 2, 3, 1, 6, 1, 0, 3, 2, 3, 3, 1, 2, 3, 4, 1, 6, 1, 3, 4, 2, 1, 5, 1, 2, 3, 3, 1, 4, 3, 4, 3, 2, 1, 9, 1, 2, 4, 0, 3, 6, 1, 3, 3, 6, 1, 4, 1, 2, 4, 3, 3, 6, 1, 5, 2, 2, 1, 9, 3, 2, 3, 4, 1, 8, 3, 3

Offset: 1

Author

Wesley Ivan Hurt, Jun 11 2021

Keywords

• nonn
• easy

Examples

			a(24) = 4; The divisors d of 24 are {1, 2, 3, 4, 6, 8, 12, 24} with corresponding values of sigma(d) {1, 3, 4, 7, 12, 15, 28, 60}. There are 4 even values of sigma(d).
		

Links

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

Crossrefs

Cf. A000005, A000203 (sigma), A065704.

Programs

Mathematica

Table[Sum[Mod[DivisorSigma[1, k] + 1, 2] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
f1[p_, e_] := e+1; f2[p_, e_] := If[p == 2, e+1, Floor[e/2] + 1]; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 06 2023 *)

PARI

a(n) = sumdiv(n, d, !(sigma(d) % 2)); \\ Michel Marcus, Jun 11 2021

Formula

a(n) = Sum_{d|n} ((sigma(d)+1) mod 2).

a(n) = A000005(n) - A065704(n). - Antti Karttunen, Dec 13 2021