A073705 a(n) = Sum_{ d divides n } (n/d)^(2d).

1, 5, 10, 33, 26, 182, 50, 577, 811, 1750, 122, 16194, 170, 18982, 74900, 135425, 290, 847127, 362, 2498178, 4901060, 4209430, 530, 78564226, 9766251, 67138102, 387952660, 542674914, 842, 4866184552, 962, 8606778369, 31382832260, 17179953862, 6385992100, 422091411267, 1370, 274878038710

Offset: 1

Author

Paul D. Hanna, Aug 04 2002

Keywords

• easy
• nonn

Comments

a(n) is the number of linear partitions of the linearly ordered set [n] = {1,2,...,n} with blocks of the same size, where each block has two element marked (possibly equal). For instance, for n = 3, we have the following 10 linear partitions (where the marked elements are denoted by a and b, or by X when they coincide):

(X)(X)(X), (ab3), (a2b), (1ab), (ba3), (b2a), (1ba), (X23), (1X3), (12X). - Emanuele Munarini, Feb 03 2014

Examples

			a(10) = (10/1)^(2*1) +(10/2)^(2*2) +(10/5)^(2*5) +(10/10)^(2*10) = 1750 because positive divisors of 10 are 1, 2, 5, 10.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..3143

Crossrefs

Cf. A038041, A055225, A151954, A236696.

Programs

Mathematica

Table[Total[Quotient[n, x = Divisors[n]]^(2*x)], {n, 34}] (* Jayanta Basu, Jul 08 2013 *)

Maxima

a(n):= lsum(d^(2*n/d),d,listify(divisors(n)));
makelist(a(n),n,1,40); /* Emanuele Munarini , Feb 03 2014 */

PARI

a(n)=sumdiv(n, d, (d)^(2*n/d) );  /* Joerg Arndt, Oct 07 2012 */

Formula

G.f.: Sum_{n>=1} -log(1 - (n^2)*x^n)/n = Sum_{n>=1} a(n) x^n/n.

G.f.: Sum_{k>=1} k^2*x^k/(1-k^2*x^k). - Benoit Cloitre, Apr 21 2003

Extensions

Corrected a(14) and inserted missing a(16) by Jayanta Basu, Jul 08 2013