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-/
import FormalConjecturesUtil
namespace OeisA51903Maximum exponent in the prime factorization of $n$.
def a (n : ℕ) : ℕ :=
(n.primeFactorsList.map (n.primeFactorsList.count ·)).foldr max 0@[category test, AMS 11]
theorem a_1 : a 1 = 0 := ⊢ a 1 = 0
All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 1 := ⊢ a 2 = 1
All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 1 := ⊢ a 3 = 1
All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 2 := ⊢ a 4 = 2
All goals completed! 🐙@[category test, AMS 11]
theorem a_5 : a 5 = 1 := ⊢ a 5 = 1
All goals completed! 🐙Are there composite numbers $n > 4$ such that $n \equiv a(n) \pmod{\phi(n)}$?
Thomas Ordowski, Dec 02 2019
@[category research open, AMS 11]
theorem conjecture1 :
answer(sorry) ↔ ∃ n : ℕ, 4 < n ∧ ¬ n.Prime ∧ n.totient ∣ (n - a n) := ⊢ True ↔ ∃ n, 4 < n ∧ ¬Nat.Prime n ∧ n.totient ∣ n - a n
All goals completed! 🐙Are there odd numbers $n$ such that $a(n) > 1$ and $n \equiv a(n) \pmod{\lambda(n)}$? (Equivalently, odd numbers $n$ such that $a(n) > 1$ and $b^n \equiv b^{a(n)} \pmod n$ for all $b$.)
Thomas Ordowski, Dec 02 2019
@[category research open, AMS 11]
theorem conjecture2 :
answer(sorry) ↔ ∃ n : ℕ, Odd n ∧ 1 < a n ∧ ∀ b : ℕ, b ^ n ≡ b ^ (a n) [MOD n] := ⊢ True ↔ ∃ n, Odd n ∧ 1 < a n ∧ ∀ (b : ℕ), b ^ n ≡ b ^ a n [MOD n]
All goals completed! 🐙Are there odd numbers $n$ such that $a(n) > 1$ and $n \equiv a(n) \pmod{\operatorname{ord}_n(2)}$? (Equivalently, odd numbers $n$ such that $a(n) > 1$ and $2^n \equiv 2^{a(n)} \pmod n$.)
Thomas Ordowski, Dec 02 2019
@[category research open, AMS 11]
theorem conjecture3 :
answer(sorry) ↔ ∃ n : ℕ, Odd n ∧ 1 < a n ∧ 2 ^ n ≡ 2 ^ (a n) [MOD n] := ⊢ True ↔ ∃ n, Odd n ∧ 1 < a n ∧ 2 ^ n ≡ 2 ^ a n [MOD n]
All goals completed! 🐙end OeisA51903