This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A113258 #13 Jun 08 2025 03:18:35
%S A113258 1,3,11,125,16824569,1329227995784915877642188398793079569
%N A113258 Ascending descending base exponent transform of factorials.
%C A113258 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?
%F A113258 a(n) = Sum_{i = 1..n} (i!)^((n-i+1)!).
%F A113258 a(n) = Sum_{i = 1..n} (n-i+1)!^i!.
%F A113258 a(n) = Sum_{i = 1..n} (A000142(i))^(A000142(n-i+1)).
%F A113258 a(n) ~ 2^((n-1)!). - _Vaclav Kotesovec_, Jun 08 2025
%e A113258 a(1) = 1 because (1!)^(1!) = 1^1 = 1.
%e A113258 a(2) = 3 because (1!)^(2!) + (2!)^(1!) = 1 + 2 = 3.
%e A113258 a(3) = 11 = (1!)^(3!) + (2!)^(2!) + (3!)^(1!) = 1^6 + 2^2 + 6^1 = 11.
%e A113258 a(4) = 125 = (1!)^(4!) + (2!)^(3!) + (3!)^(2!) + (4!)^(1!).
%e A113258 a(6) = 1329227995784915877642188398793079569 = 1^720 + 2^120 + 6^24 + 24^6 + 120^2 + 720^1.
%e A113258 a(7) = 1!^7! + 2!^6! + 3!^5! + 4!^4! + 5!^3! + 6!^2! + 7!^1! has 217 digits.
%t A113258 Table[Sum[((k)!)^(n - k + 1)!, {k, 1, n}], {n,1,5}] (* _G. C. Greubel_, May 18 2017 *)
%o A113258 (PARI) for(n=1,5, print1(sum(k=1,n, (k!)^((n-k+1)!)), ", ")) \\ _G. C. Greubel_, May 18 2017
%Y A113258 Cf. A000142, A005408, A113122, A113153, A113154.
%K A113258 nonn,easy
%O A113258 1,2
%A A113258 _Jonathan Vos Post_, Jan 07 2006