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you may not use this file except in compliance with the License.
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import FormalConjecturesUtilNo powers as partition numbers
There are no partition numbers $a(k)$ of the form $x^m$, with $x,m$ integers $>1$.
namespace OeisA41
open Nat
The n-th partition number.
def a (n : ℕ) : ℕ := Fintype.card (Nat.Partition n)
@[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 = 1 := ⊢ a 1 = 1 All goals completed! 🐙
@[category test, AMS 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙
@[category test, AMS 11]
theorem a_3 : a 3 = 3 := ⊢ a 3 = 3 All goals completed! 🐙
@[category test, AMS 11]
theorem a_4 : a 4 = 5 := ⊢ a 4 = 5 All goals completed! 🐙
@[category test, AMS 11]
theorem a_5 : a 5 = 7 := ⊢ a 5 = 7 All goals completed! 🐙
There are no partition numbers $a(k)$ of the form $x^m$, with $x,m$ integers $>1$. See comment by Zhi-Wei Sun (Dec 02 2013).
@[category research open, AMS 11]
theorem noPowerPartitionNumber : answer(sorry) ↔ ∀ k, ¬IsPerfectPower (a k) := ⊢ True ↔ ∀ (k : ℕ), ¬(a k).IsPerfectPower
All goals completed! 🐙
end OeisA41