A152211 a(n) = n * sigma_0(n) + sigma_1(n).

2, 7, 10, 19, 16, 36, 22, 47, 40, 58, 34, 100, 40, 80, 84, 111, 52, 147, 58, 162, 116, 124, 70, 252, 106, 146, 148, 224, 88, 312, 94, 255, 180, 190, 188, 415, 112, 212, 212, 410, 124, 432, 130, 348, 348, 256, 142, 604

Offset: 1

Author

Ctibor O. Zizka, Nov 29 2008

Keywords

• easy
• nonn

Comments

a(n) is the sum of all parts plus the total number of parts of all partitions of n into equal parts. - Omar E. Pol, Nov 30 2019

Examples

			For n = 4 the partitions of 4 into equal parts are [4], [2,2], [1,1,1,1]. The sum of all parts is 4 + 2 + 2 + 1 + 1 + 1 + 1 = 12. There are 7 parts, so a(4) = 12 + 7 = 19. - _Omar E. Pol_, Nov 30 2019
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000203, A038040, A094471, A244051.

Cf. A001620, A013661.

Programs

Mathematica

Array[Total[{#, 1} DivisorSigma[{0, 1}, #]] &, 48] (* Michael De Vlieger, Dec 01 2019 *)

PARI

a(n) =  n*numdiv(n) + sigma(n) \\ Michel Marcus, Jun 02 2013

Formula

a(n) = n * A000005(n) + A000203(n).

a(n) = A038040(n) + A000203(n). - Torlach Rush, Feb 01 2019

G.f.: Sum_{k>=1} (k + 1) * x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Aug 14 2019

Sum_{k=1..n} a(k) ~ (n^2/2) * (log(n) + 2*gamma + zeta(2) - 1/2), where gamma is Euler's constant (A001620). - Amiram Eldar, Feb 01 2025