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import FormalConjecturesUtil
namespace OeisA113258open Nat
The primary defining sequence a.
$a(n)$ is the ascending descending base exponent transform of factorials.
$$a(n) = \sum_{i = 1}^n (i!) ^ {(n-i+1)!}$$
def a (n : ℕ) : ℕ :=
∑ i ∈ Finset.range n, (i + 1)! ^ (n - i)!@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 3 := ⊢ a 2 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 11 := ⊢ a 3 = 11 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 125 := ⊢ a 4 = 125 All goals completed! 🐙Is there a nontrivial power after $a(4) = 5^3$?
@[category research open, AMS 11]
theorem conjecture :
answer(sorry) ↔ ∃ n > 4, ∃ b > 1, ∃ e > 1, a n = b ^ e := ⊢ True ↔ ∃ n > 4, ∃ b > 1, ∃ e > 1, a n = b ^ e
All goals completed! 🐙end OeisA113258