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import FormalConjecturesUtil
namespace OeisA113271
The primary defining sequence a.
$a(n)$ is the ascending descending base exponent transform of $2^n$.
$$a(n) = \sum_{i=0}^n 2^{i \cdot 2^{n-i}}$$
def a (n : ℕ) : ℕ :=
∑ i ∈ Finset.range (n + 1), 2 ^ (i * 2 ^ (n - i))@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 3 := ⊢ a 1 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 9 := ⊢ a 2 = 9 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 41 := ⊢ a 3 = 41 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 593 := ⊢ a 4 = 593 All goals completed! 🐙The smallest primes in this (always odd) sequence are $a(1) = 3$, $a(3) = 41$ and $a(5) = 543$. What is the next prime?
@[category research open, AMS 11]
theorem conjecture1 :
answer(sorry) = a (sInf {n : ℕ | 5 < n ∧ (a n).Prime}) := ⊢ sorry = a (sInf {n | 5 < n ∧ Nat.Prime (a n)})
All goals completed! 🐙end OeisA113271