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module
public import FormalConjecturesUtilNo powers as partition numbers
There are no partition numbers $a(k)$ of the form $x^m$, with $x,m$ integers $>1$.
Reference: A41
@[expose] public sectionnamespace OeisA41open 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! 🐙--TODO: these were previously proven with `native_decide`,
-- but really, one should rove the recurrence relation and use that
@[category test, AMS 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2
All goals completed! 🐙 --this was previously proven using `native_decide`@[category test, AMS 11]
theorem a_3 : a 3 = 3 := ⊢ a 3 = 3
All goals completed! 🐙 --this was previously proven using `native_decide`@[category test, AMS 11]
theorem a_4 : a 4 = 5 := ⊢ a 4 = 5
All goals completed! 🐙 --this was previously proven using `native_decide`@[category test, AMS 11]
theorem a_5 : a 5 = 7 := ⊢ a 5 = 7
All goals completed! 🐙 --this was previously proven using `native_decide`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