A055225 a(n) = Sum_{k divides n} (n/k)^k.

1, 3, 4, 9, 6, 24, 8, 41, 37, 68, 12, 258, 14, 192, 384, 593, 18, 1557, 20, 2794, 2552, 2192, 24, 16730, 3151, 8388, 20440, 35394, 30, 116474, 32, 135457, 178512, 131396, 94968, 1111035, 38, 524688, 1596560, 2530986, 42, 7280934, 44, 8403778

Offset

1,2

Comments

a(n) is the number of (nonempty) linear partitions of the linearly ordered set [n] = {1,2,...,n} with blocks of the same size, where each block has exactly one element marked. For instance, for n = 4, we have the following 9 linear partitions (where the marked elements are denoted by *):

. (*)(*)(*)(*), (*2)(*4), (*234),

. (*2)(3*), (1*34),

. (1*)(*4), (12*4),

. (1*)(3*), (123*).

- Emanuele Munarini, Feb 03 2014

Links

Nick Hobson, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{n>=1} -log(1 - n*x^n)/n = Sum_{n>=0} a(n) x^n/n. - Paul D. Hanna, Aug 04 2002

G.f.: Sum_{n>0} n*x^n/(1-n*x^n). - Vladeta Jovovic, Sep 02 2002

Sum_{k=1..n} a(k) ~ 3^((n + 3 - mod(n,3))/3)/2. - Vaclav Kotesovec, Aug 07 2022

Example

a(10) = 10^1 + 5^2 + 2^5 + 1^10 = 68 because positive divisors of 10 are 1, 2, 5, 10.

Mathematica

Table[Total[Quotient[n, x = Divisors[n]]^x], {n, 44}] (* Jayanta Basu, Jul 08 2013 *)
Table[Sum[d^(n/d), {d, Divisors[n]}], {n, 1, 100}] (* Emanuele Munarini, Feb 03 2014 *)

Prog

(PARI) vector(44, n, sumdiv(n, d, (n/d)^d))
(PARI) a(n) = sumdiv(n, d, d^(n/d) ); \\ Joerg Arndt, Apr 14 2013
(Maxima) a(n) := lsum(d^(n/d), d, listify(divisors(n))); makelist(a(n), n, 1, 40); /* Emanuele Munarini, Feb 03 2014 */

Crossrefs

Cf. A005225, A038041, A236696.

Sequence in context: A157020 A180253 A264786 * A054791 A167531 A344195

Adjacent sequences: A055222 A055223 A055224 * A055226 A055227 A055228

Keyword

nonn

Author

Leroy Quet, Jun 20 2000

Extensions

More terms from James A. Sellers, Jul 04 2000

Duplicate g.f. removed by Franklin T. Adams-Watters, Sep 01 2009

Status

approved