A138564 a(1) = 1; a(n) = a(n-1) + (n!)^3.

1, 9, 225, 14049, 1742049, 374990049, 128399054049, 65676719822049, 47850402559694049, 47832576242431694049, 63649302669112063694049, 109966989623147836159694049, 241567605673714904675071694049

Offset: 1

Author

Jonathan Vos Post, May 18 2008

Keywords

• easy
• nonn

Comments

By sum of cubes factorization, every a(n) > 1 is a multiple of 9, hence none of these are prime, unlike the case of sum of squares of factorials (i.e. (1!)^2 + (2!)^2+ (3!)^2+ (4!)^2 = 617 is prime; 41117342095090841723228045851817 = (1!)^2 + (2!)^2 + (3!)^2 + (4!)^2 + (5!)^2 + (6!)^2 + (7!)^2 + (8!)^2 + (9!)^2 + (10!)^2 + (11!)^2 + (12!)^2 + (13!)^2 + (14!)^2 + (15!)^2 + (16!)^2 + (17!)^2 + (18!)^2 is prime).

Examples

			a(18) = (1!)^3 + (2!)^3 + (3!)^3 + (4!)^3 + (5!)^3 + (6!)^3 + (7!)^3 + (8!)^3 + (9!)^3 + (10!)^3 + (11!)^3 + (12!)^3 + (13!)^3 + (14!)^3 + (15!)^3 + (16!)^3 + (17!)^3 + (18!)^3 = 262480797594664584673157017306412926841599694049.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..181

Crossrefs

Cf. A000142, A000578, A100288.

Sum_{k=1..n} (k!)^m: A007489 (m=1), A104344 (m=2), this sequence (m=3), A289945 (m=4), A289946 (m=6).

Programs

Mathematica

nxt[{n_,a_}]:={n+1,a+((n+1)!)^3}; Transpose[NestList[nxt,{1,1},20]][[2]] (* Harvey P. Dale, Mar 08 2015 *)

Formula

a(n) = Sum_{k=1..n} (k!)^3 = Sum_{k=1..n} A000578(A000142(n)).