A113258 Ascending descending base exponent transform of factorials.

1, 3, 11, 125, 16824569, 1329227995784915877642188398793079569

Offset: 1

Author

Jonathan Vos Post, Jan 07 2006

Keywords

• nonn
• easy

Comments

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The smallest primes in this (always odd) sequence are a(2) = 3 and a(3) = 11. What is the next prime? Is there a nontrivial power after a(4) = 5^3?

Examples

			a(1) = 1 because (1!)^(1!) = 1^1 = 1.
a(2) = 3 because (1!)^(2!) + (2!)^(1!) = 1 + 2 = 3.
a(3) = 11 = (1!)^(3!) + (2!)^(2!) + (3!)^(1!) = 1^6 + 2^2 + 6^1 = 11.
a(4) = 125 = (1!)^(4!) + (2!)^(3!) + (3!)^(2!) + (4!)^(1!).
a(6) = 1329227995784915877642188398793079569 = 1^720 + 2^120 + 6^24 + 24^6 + 120^2 + 720^1.
a(7) = 1!^7! + 2!^6! + 3!^5! + 4!^4! + 5!^3! + 6!^2! + 7!^1! has 217 digits.
		

Crossrefs

Cf. A000142, A005408, A113122, A113153, A113154.

Programs

Mathematica

Table[Sum[((k)!)^(n - k + 1)!, {k, 1, n}], {n,1,5}] (* G. C. Greubel, May 18 2017 *)

PARI

for(n=1,5, print1(sum(k=1,n, (k!)^((n-k+1)!)), ", ")) \\ G. C. Greubel, May 18 2017

Formula

a(n) = Sum_{i = 1..n} (i!)^((n-i+1)!).

a(n) = Sum_{i = 1..n} (n-i+1)!^i!.

a(n) = Sum_{i = 1..n} (A000142(i))^(A000142(n-i+1)).

a(n) ~ 2^((n-1)!). - Vaclav Kotesovec, Jun 08 2025