FormalConjectures.OEIS.«108569»

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import FormalConjecturesUtil

Numbers $n$ such that $\phi(n) = \phi(n + \phi(n))$

References:

A108569

namespace OeisA108569open scoped Nat

The predicate defining whether $k$ belongs to the sequence.

def A (k : ℕ) : Prop := 0 < k ∧ φ k = φ (k + φ k)

instance : DecidablePred A := by⊢ DecidablePred A
  unfold A⊢ DecidablePred fun k ↦ 0 < k ∧ φ k = φ (k + φ k)
  infer_instanceAll goals completed! 🐙

The primary defining sequence a. a n is the $(n+1)$-th positive integer $k$ such that $\phi(k) = \phi(k + \phi(k))$.

noncomputable def a (n : ℕ) : ℕ :=
  n.nth A

Term theorems verifying the first few values of the sequence against the official OEIS b-file


@[category test, AMS 11]
theorem a_0 : a 0 = 1 := by⊢ a 0 = 1
  have h1 : A 1 := by decideh1:A 1⊢ a 0 = 1h1:A 1⊢ a 0 = 1
  have hcnt : Nat.count A 1 = 0 := by decideh1:A 1hcnt:Nat.count A 1 = 0⊢ a 0 = 1h1:A 1hcnt:Nat.count A 1 = 0⊢ a 0 = 1
  have := Nat.nth_count (p := A) h1h1:A 1hcnt:Nat.count A 1 = 0this:Nat.nth A (Nat.count A 1) = 1⊢ a 0 = 1
  rwa [hcnth1:A 1hcnt:Nat.count A 1 = 0this:Nat.nth A 0 = 1⊢ a 0 = 1]h1:A 1hcnt:Nat.count A 1 = 0this:Nat.nth A 0 = 1⊢ a 0 = 1 at this



@[category test, AMS 11]
theorem a_1 : a 1 = 4 := by⊢ a 1 = 4
  have h4 : A 4 := by decideh4:A 4⊢ a 1 = 4h4:A 4⊢ a 1 = 4
  have hcnt : Nat.count A 4 = 1 := by decideh4:A 4hcnt:Nat.count A 4 = 1⊢ a 1 = 4h4:A 4hcnt:Nat.count A 4 = 1⊢ a 1 = 4
  have := Nat.nth_count (p := A) h4h4:A 4hcnt:Nat.count A 4 = 1this:Nat.nth A (Nat.count A 4) = 4⊢ a 1 = 4
  rwa [hcnth4:A 4hcnt:Nat.count A 4 = 1this:Nat.nth A 1 = 4⊢ a 1 = 4]h4:A 4hcnt:Nat.count A 4 = 1this:Nat.nth A 1 = 4⊢ a 1 = 4 at this



@[category test, AMS 11]
theorem a_2 : a 2 = 8 := by⊢ a 2 = 8
  have h8 : A 8 := by decideh8:A 8⊢ a 2 = 8h8:A 8⊢ a 2 = 8
  have hcnt : Nat.count A 8 = 2 := by decideh8:A 8hcnt:Nat.count A 8 = 2⊢ a 2 = 8h8:A 8hcnt:Nat.count A 8 = 2⊢ a 2 = 8
  have := Nat.nth_count (p := A) h8h8:A 8hcnt:Nat.count A 8 = 2this:Nat.nth A (Nat.count A 8) = 8⊢ a 2 = 8
  rwa [hcnth8:A 8hcnt:Nat.count A 8 = 2this:Nat.nth A 2 = 8⊢ a 2 = 8]h8:A 8hcnt:Nat.count A 8 = 2this:Nat.nth A 2 = 8⊢ a 2 = 8 at this



@[category test, AMS 11]
theorem a_3 : a 3 = 16 := by⊢ a 3 = 16
  have h16 : A 16 := by decideh16:A 16⊢ a 3 = 16h16:A 16⊢ a 3 = 16
  have hcnt : Nat.count A 16 = 3 := by decideh16:A 16hcnt:Nat.count A 16 = 3⊢ a 3 = 16h16:A 16hcnt:Nat.count A 16 = 3⊢ a 3 = 16
  have := Nat.nth_count (p := A) h16h16:A 16hcnt:Nat.count A 16 = 3this:Nat.nth A (Nat.count A 16) = 16⊢ a 3 = 16
  rwa [hcnth16:A 16hcnt:Nat.count A 16 = 3this:Nat.nth A 3 = 16⊢ a 3 = 16]h16:A 16hcnt:Nat.count A 16 = 3this:Nat.nth A 3 = 16⊢ a 3 = 16 at this



@[category test, AMS 11]
theorem a_4 : a 4 = 32 := by⊢ a 4 = 32
  have h32 : A 32 := by decideh32:A 32⊢ a 4 = 32h32:A 32⊢ a 4 = 32
  have hcnt : Nat.count A 32 = 4 := by decideh32:A 32hcnt:Nat.count A 32 = 4⊢ a 4 = 32h32:A 32hcnt:Nat.count A 32 = 4⊢ a 4 = 32
  have := Nat.nth_count (p := A) h32h32:A 32hcnt:Nat.count A 32 = 4this:Nat.nth A (Nat.count A 32) = 32⊢ a 4 = 32
  rwa [hcnth32:A 32hcnt:Nat.count A 32 = 4this:Nat.nth A 4 = 32⊢ a 4 = 32]h32:A 32hcnt:Nat.count A 32 = 4this:Nat.nth A 4 = 32⊢ a 4 = 32 at this

Conjecture: Except for the first term all terms are even.

@[category research open, AMS 11]
theorem conjecture : ∀ n, 0 < n → Even (a n) := by⊢ ∀ (n : ℕ), 0 < n → Even (OeisA108569.a n)
  sorryAll goals completed! 🐙

end OeisA108569