FormalConjectures.OEIS.«113019»

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import FormalConjecturesUtil

Number of digits of n raised to the power of the digital root of n

References:

A113019

namespace OeisA113019open Nat

a n is the (Number of digits of n) raised to the power of (the digital root of n), with appropriate adjustments for $n=0$.

def a (n : ℕ) : ℕ :=
  -- The base: number of digits of n (adjusting n=0 to have 1 digit, like n=1).
  let numDigits : ℕ := (Nat.digits 10 (max 1 n)).length

  -- The exponent: digital root of n. This correctly yields 0 for n=0,
  -- and the standard 1..9 for n>0.
  let digitalRoot : ℕ := if n = 0 then 0 else (n - 1) % 9 + 1

  numDigits ^ digitalRoot

@[category test, AMS 11]
theorem a_0 : a 0 = 1 := by⊢ a 0 = 1 native_decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_1 : a 1 = 1 := by⊢ a 1 = 1 native_decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_2 : a 2 = 1 := by⊢ a 2 = 1 native_decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_3 : a 3 = 1 := by⊢ a 3 = 1 native_decideAll goals completed! 🐙

@[category test, AMS 11]
theorem a_4 : a 4 = 1 := by⊢ a 4 = 1 native_decideAll goals completed! 🐙

$n=1$ and $32$ are fixed points. Are there any others?

Yes: 9^9 = 387420489 is also a fixed point. - Kenta Kitamura, Aug 14 2026


@[category research solved, AMS 11]
theorem conjecture : answer(False) ↔ ∀ n : ℕ, a n = n → n = 1 ∨ n = 32 := by⊢ False ↔ ∀ (n : ℕ), a n = n → n = 1 ∨ n = 32
  change False ↔ ∀ n : ℕ, a n = n → n = 1 ∨ n = 32⊢ False ↔ ∀ (n : ℕ), a n = n → n = 1 ∨ n = 32
  constructormp⊢ False → ∀ (n : ℕ), OeisA113019.a n = n → n = 1 ∨ n = 32mpr⊢ (∀ (n : ℕ), a n = n → n = 1 ∨ n = 32) → False
  ·mp⊢ False → ∀ (n : ℕ), OeisA113019.a n = n → n = 1 ∨ n = 32 exact False.elimAll goals completed! 🐙
  ·mpr⊢ (∀ (n : ℕ), a n = n → n = 1 ∨ n = 32) → False intro hmprh:∀ (n : ℕ), a n = n → n = 1 ∨ n = 32⊢ False
    have hfixed : a 387420489 = 387420489 := by⊢ False ↔ ∀ (n : ℕ), a n = n → n = 1 ∨ n = 32mprh:∀ (n : ℕ), a n = n → n = 1 ∨ n = 32hfixed:a 387420489 = 387420489⊢ False
      simp [a]mprh:∀ (n : ℕ), a n = n → n = 1 ∨ n = 32hfixed:a 387420489 = 387420489⊢ Falsemprh:∀ (n : ℕ), a n = n → n = 1 ∨ n = 32hfixed:a 387420489 = 387420489⊢ False
    rcases h 387420489 hfixed with h | hmpr.inlh✝:∀ (n : ℕ), a n = n → n = 1 ∨ n = 32hfixed:a 387420489 = 387420489h:387420489 = 1⊢ Falsempr.inrh✝:∀ (n : ℕ), a n = n → n = 1 ∨ n = 32hfixed:a 387420489 = 387420489h:387420489 = 32⊢ False <;>mpr.inlh✝:∀ (n : ℕ), a n = n → n = 1 ∨ n = 32hfixed:a 387420489 = 387420489h:387420489 = 1⊢ Falsempr.inrh✝:∀ (n : ℕ), a n = n → n = 1 ∨ n = 32hfixed:a 387420489 = 387420489h:387420489 = 32⊢ False omegaAll goals completed! 🐙

end OeisA113019