1. FormalConjectures
  2. OEIS
  3. «67599»
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Decimal encoding of the prime factorization of $n$

If $n$ has prime factorization $p_1^{e_1} \dots p_r^{e_r}$ with $p_1 < \dots < p_r$, then its decimal encoding is $p_1 e_1 \dots p_r e_r$.

References:

namespace OeisA67599

Concatenates two natural numbers $a$ and $b$ base 10.

def concatenateNats (a b : ℕ) : ℕ := a * (10 ^ (Nat.digits 10 b).length) + b

Decimal encoding of the prime factorization of $n$.

def a (n : ℕ) : ℕ := if n < 2 then 0 else let factors : List ℕ := n.primeFactorsList.dedup let flat_list : List ℕ := factors.flatMap fun p ↦ [p, n.primeFactorsList.count p] flat_list.foldl concatenateNats 0@[category test, AMS 11] theorem a_2 : a 2 = 21 := ⊢ a 2 = 21 All goals completed! 🐙@[category test, AMS 11] theorem a_3 : a 3 = 31 := ⊢ a 3 = 31 All goals completed! 🐙@[category test, AMS 11] theorem a_4 : a 4 = 22 := ⊢ a 4 = 22 All goals completed! 🐙@[category test, AMS 11] theorem a_5 : a 5 = 51 := ⊢ a 5 = 51 All goals completed! 🐙@[category test, AMS 11] theorem a_6 : a 6 = 2131 := ⊢ a 6 = 2131 All goals completed! 🐙

"$a(31) = a(177147) = 311$. Is there any solution to $a(n) = n$?

    Franklin T. Adams-Watters, Dec 18 2006"

@[category research open, AMS 11] theorem conjecture : answer(sorry) ↔ ∃ n : ℕ, 2 ≤ n ∧ a n = n := ⊢ True ↔ ∃ n, 2 ≤ n ∧ a n = n All goals completed! 🐙end OeisA67599