FormalConjectures.OEIS.«110475»

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import FormalConjecturesUtil

Number of symbols '*' and '^' to write the canonical prime factorization of n

The canonical prime factorization is $n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$. The written form is $p_1^{\wedge} e_1 * p_2^{\wedge} e_2 * \cdots * p_k^{\wedge} e_k$, where the $\wedge$ appears only if $e_i > 1$. $a(n) = (\text{number of distinct prime factors}) - 1 +$ $(\text{number of distinct prime factors with exponent } > 1)$.

References:

A110475

namespace OeisA110475

The primary defining sequence a. $a(n)$ is the number of symbols '*' and '^' to write the canonical prime factorization of $n$.

noncomputable def a (n : ℕ) : ℕ :=
  let f := Nat.factorization n
  let s := f.support
  let numDistinctPrimes := s.card
  let numAsterisks := numDistinctPrimes - 1
  let numCarets := (s.filter fun p => f p > 1).card
  numAsterisks + numCarets

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

@[category test, AMS 11]
theorem a_1 : a 1 = 0 := by⊢ a 1 = 0
  dsimp [a]⊢ 0 + {p ∈ ∅ | (Nat.factorization 1) p > 1}.card = 0
  simpAll goals completed! 🐙



@[category test, AMS 11]
theorem a_2 : a 2 = 0 := by⊢ a 2 = 0
  dsimp [a]⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (Nat.factorization 2) p > 1}.card = 0
  have h2 : Nat.Prime 2 := by⊢ a 2 = 0h2:Nat.Prime 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (Nat.factorization 2) p > 1}.card = 0 decideh2:Nat.Prime 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (Nat.factorization 2) p > 1}.card = 0h2:Nat.Prime 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (Nat.factorization 2) p > 1}.card = 0
  rw [Nat.Prime.factorization h2h2:Nat.Prime 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (fun₀ | 2 => 1) p > 1}.card = 0h2:Nat.Prime 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (fun₀ | 2 => 1) p > 1}.card = 0]h2:Nat.Prime 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (fun₀ | 2 => 1) p > 1}.card = 0
  simpAll goals completed! 🐙



@[category test, AMS 11]
theorem a_3 : a 3 = 0 := by⊢ a 3 = 0
  dsimp [a]⊢ (insert 3 ∅).card - 1 + {p ∈ insert 3 ∅ | (Nat.factorization 3) p > 1}.card = 0
  have h3 : Nat.Prime 3 := by⊢ a 3 = 0h3:Nat.Prime 3⊢ (insert 3 ∅).card - 1 + {p ∈ insert 3 ∅ | (Nat.factorization 3) p > 1}.card = 0 decideh3:Nat.Prime 3⊢ (insert 3 ∅).card - 1 + {p ∈ insert 3 ∅ | (Nat.factorization 3) p > 1}.card = 0h3:Nat.Prime 3⊢ (insert 3 ∅).card - 1 + {p ∈ insert 3 ∅ | (Nat.factorization 3) p > 1}.card = 0
  rw [Nat.Prime.factorization h3h3:Nat.Prime 3⊢ (insert 3 ∅).card - 1 + {p ∈ insert 3 ∅ | (fun₀ | 3 => 1) p > 1}.card = 0h3:Nat.Prime 3⊢ (insert 3 ∅).card - 1 + {p ∈ insert 3 ∅ | (fun₀ | 3 => 1) p > 1}.card = 0]h3:Nat.Prime 3⊢ (insert 3 ∅).card - 1 + {p ∈ insert 3 ∅ | (fun₀ | 3 => 1) p > 1}.card = 0
  simpAll goals completed! 🐙



@[category test, AMS 11]
theorem a_4 : a 4 = 1 := by⊢ a 4 = 1
  dsimp [a]⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (Nat.factorization 4) p > 1}.card = 1
  have : (4 : ℕ) = 2 ^ 2 := by⊢ a 4 = 1this:4 = 2 ^ 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (Nat.factorization 4) p > 1}.card = 1 rflthis:4 = 2 ^ 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (Nat.factorization 4) p > 1}.card = 1this:4 = 2 ^ 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (Nat.factorization 4) p > 1}.card = 1
  rw [this,this:4 = 2 ^ 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (2 ^ 2).factorization p > 1}.card = 1this:4 = 2 ^ 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (fun₀ | 2 => 2) p > 1}.card = 1 Nat.Prime.factorization_pow (bythis:4 = 2 ^ 2⊢ Nat.Prime 2this:4 = 2 ^ 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (fun₀ | 2 => 2) p > 1}.card = 1 decideAll goals completed! 🐙this:4 = 2 ^ 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (fun₀ | 2 => 2) p > 1}.card = 1)]this:4 = 2 ^ 2⊢ (insert 2 ∅).card - 1 + {p ∈ insert 2 ∅ | (fun₀ | 2 => 2) p > 1}.card = 1
  simp [Finset.filter_singleton]All goals completed! 🐙



@[category test, AMS 11]
theorem a_5 : a 5 = 0 := by⊢ a 5 = 0
  dsimp [a]⊢ (insert 5 ∅).card - 1 + {p ∈ insert 5 ∅ | (Nat.factorization 5) p > 1}.card = 0
  have h5 : Nat.Prime 5 := by⊢ a 5 = 0h5:Nat.Prime 5⊢ (insert 5 ∅).card - 1 + {p ∈ insert 5 ∅ | (Nat.factorization 5) p > 1}.card = 0 decideh5:Nat.Prime 5⊢ (insert 5 ∅).card - 1 + {p ∈ insert 5 ∅ | (Nat.factorization 5) p > 1}.card = 0h5:Nat.Prime 5⊢ (insert 5 ∅).card - 1 + {p ∈ insert 5 ∅ | (Nat.factorization 5) p > 1}.card = 0
  rw [Nat.Prime.factorization h5h5:Nat.Prime 5⊢ (insert 5 ∅).card - 1 + {p ∈ insert 5 ∅ | (fun₀ | 5 => 1) p > 1}.card = 0h5:Nat.Prime 5⊢ (insert 5 ∅).card - 1 + {p ∈ insert 5 ∅ | (fun₀ | 5 => 1) p > 1}.card = 0]h5:Nat.Prime 5⊢ (insert 5 ∅).card - 1 + {p ∈ insert 5 ∅ | (fun₀ | 5 => 1) p > 1}.card = 0
  simpAll goals completed! 🐙

The set of exceptional integers.

def exceptionalSet : Finset ℕ :=
  {1, 2, 3, 4, 5, 6, 7, 9, 11}

It is conjectured that $1,2,3,4,5,6,7,9,11$ are the only positive integers which cannot be represented as the sum of two elements of indices $n$ such that $a(n) = 1$.

@[category research open, AMS 11]
theorem conjecture :
  ∀ m > 0, m ∉ exceptionalSet ↔ ∃ x y : ℕ, a x = 1 ∧ a y = 1 ∧ m = x + y := by⊢ ∀ m > 0, m ∉ exceptionalSet ↔ ∃ x y, OeisA110475.a x = 1 ∧ OeisA110475.a y = 1 ∧ m = x + y
  sorryAll goals completed! 🐙

end OeisA110475