A057625 a(n) = n! * sum 1/k! where the sum is over all positive integers k that divide n.

1, 3, 7, 37, 121, 1201, 5041, 62161, 423361, 5473441, 39916801, 818959681, 6227020801, 130784734081, 1536517382401, 32256486662401, 355687428096001, 10679532671808001, 121645100408832001, 3770998783116364801, 59616236292028416001, 1686001119824999577601

Offset: 1

Author

Leroy Quet, Oct 09 2000

Keywords

• nonn

Comments

Sets of lists of equal size, cf. A000262. - Vladeta Jovovic, Nov 02 2003

From Gus Wiseman, Jan 10 2019: (Start)

Number of matrices whose entries are 1,...,n, up to column permutations. For example, inequivalent representatives of the a(4) = 37 matrices are:

One 1 X 4 matrix:

[1234]

12 2 X 2 matrices:

[12] [12] [13] [13] [14] [14] [23] [23] [24] [24] [34] [34]

[34] [43] [24] [42] [23] [32] [14] [41] [13] [31] [12] [21]

and 24 4 X 1 matrices:

[1][1][1][1][1][1][2][2][2][2][2][2][3][3][3][3][3][3][4][4][4][4][4][4]

[2][2][3][3][4][4][1][1][3][3][4][4][1][1][2][2][4][4][1][1][2][2][3][3]

[3][4][2][4][2][3][3][4][1][4][1][3][2][4][1][4][1][2][2][3][1][3][1][2]

[4][3][4][2][3][2][4][3][4][1][3][1][4][2][4][1][2][1][3][2][3][1][2][1]

in total 1+12+24 = 37.

(End)

Examples

			a(4) = 4! (1 + 1/2! + 1/4!) = 24 (1 + 1/2 + 1/24) = 37.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..449

Crossrefs

Cf. A000005, A005225, A038041, A038048, A061095, A121860, A209903 (exp), A236696, A320444.

Row sums of A066387.

Programs

Mathematica

a[n_] := n! DivisorSum[n, 1/#! &]; Array[a, 22] (* Jean-François Alcover, Dec 23 2015 *)

PARI

a(n)=n! * sumdiv(n, d, 1/d! );  /* Joerg Arndt, Oct 07 2012 */

Formula

E.g.f.: Sum_{n>0} (exp(x^n)-1). - Vladeta Jovovic, Dec 30 2001

E.g.f.: Sum_{k>0} x^k/k!/(1-x^k). - Vladeta Jovovic, Oct 14 2003

Equals the logarithmic derivative of A209903. - Paul D. Hanna, Jul 26 2012