A062731 Sum of divisors of 2*n.

3, 7, 12, 15, 18, 28, 24, 31, 39, 42, 36, 60, 42, 56, 72, 63, 54, 91, 60, 90, 96, 84, 72, 124, 93, 98, 120, 120, 90, 168, 96, 127, 144, 126, 144, 195, 114, 140, 168, 186, 126, 224, 132, 180, 234, 168, 144, 252, 171, 217, 216, 210, 162, 280, 216, 248, 240, 210

Offset: 1

Author

Jason Earls, Jul 11 2001

Keywords

• easy
• nonn

Comments

a(n) is also the total number of parts in all partitions of 2*n into equal parts. - Omar E. Pol, Feb 14 2021

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from Harry J. Smith]

Crossrefs

Sigma(k*n): A000203 (k=1), A144613 (k=3), A193553 (k=4, even bisection), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), A283123 (k=9).

Cf. A008438, A074400, A182818, A239052 (odd bisection), A326124 (partial sums), A054784, A215947, A336923, A346870, A346878, A346880, A355750.

Row 2 of A319526. Column & Row 2 of A216626. Row 1 of A355927.

Shallow diagonal (2n,n) of A265652. See also A244658.

Programs

Magma

[SumOfDivisors(2*n): n in [1..70]]; // Vincenzo Librandi, Oct 31 2014

Mathematica

lst={};Do[AppendTo[lst, DivisorSigma[1, n]], {n, 2, 6!, 2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 20 2008 *)
DivisorSigma[1,2*Range[60]] (* Harvey P. Dale, Jun 08 2022 *)

MuPAD

numlib::sigma(2*n)$ n=0..81 // Zerinvary Lajos, May 13 2008

PARI

vector(66,n,sigma(2*n,1))

PARI

for (n=1, 1000, write("b062731.txt", n, " ", sigma(2*n)) ) \\ Harry J. Smith, Aug 09 2009

Formula

a(n) = A000203(2*n). - R. J. Mathar, Apr 06 2011

a(n) = A000203(n) + A054785(n). - R. J. Mathar, May 19 2020

From Vaclav Kotesovec, Aug 07 2022: (Start)

Dirichlet g.f.: zeta(s) * zeta(s-1) * (3 - 2^(1-s)).

Sum_{k=1..n} a(k) ~ 5 * Pi^2 * n^2 / 24. (End)

From Miles Wilson, Sep 30 2024: (Start)

G.f.: Sum_{k>=1} k*x^(k/gcd(k, 2))/(1 - x^(k/gcd(k, 2))).

G.f.: Sum_{k>=1} k*x^(2*k/(3 + (-1)^k))/(1 - x^(2*k/(3 + (-1)^k))). (End)

Extensions

Zero removed and offset corrected by Omar E. Pol, Jul 17 2009