1,2

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?

a(n) = SUM[from i = 1 to n] (i!)^((n-i+1)!). a(n) = SUM[from i = 1 to n] (n-i+1)!^i!. a(n) = SUM[from i = 1 to n] (A000142(i))^(A000142(n-i+1)).

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.

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

Adjacent sequences: A113255 A113256 A113257 this_sequence A113259 A113260 A113261

Sequence in context: A036930 A015047 A102847 this_sequence A113848 A088075 A088076

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 07 2006