A113258 Ascending descending base exponent transform of factorials.

1, 3, 11, 125, 16824569, 1329227995784915877642188398793079569

Offset

1,2

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?

Links

Table of n, a(n) for n=1..6.

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)).

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A015047 A339326 A102847 * A113848 A287429 A274664

Adjacent sequences: A113255 A113256 A113257 * A113259 A113260 A113261

Keyword

nonn,easy

Author

Jonathan Vos Post, Jan 07 2006

Status

approved