feat(OEIS): A087719

FormalConjectures/OEIS/87719.lean

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+/-
+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+import FormalConjectures.Util.ProblemImports
+
+open Nat
+
+/-!
+# Conjectures associated with A087719
+
+Define $\varsigma(n)$ the smallest prime factor of $n$ (`Nat.minFac`). Let $a_n$ be the least
+number such that the count of numbers $k \le a_n$ with $k > \varsigma(k)^n$ exceeds the count
+of numbers with $k \le \varsigma(k)^n$.
+
+The conjecture states that $a_n = 3^n + 3 \cdot 2^n + 6$ for $n \ge 1$.
+
+*References:* [oeis.org/A087719](https://oeis.org/A087719)
+-/
+
+namespace OeisA87719
+
+/-- Count of numbers k in {1, ..., m} where k > (minFac k)^n. -/
+def countExceeding (n m : ℕ) : ℕ :=
+  (Finset.Icc 1 m).filter (fun k => k > k.minFac ^ n) |>.card
+
+/-- Count of numbers k in {1, ..., m} where k ≤ (minFac k)^n. -/
+def countNotExceeding (n m : ℕ) : ℕ :=
+  (Finset.Icc 1 m).filter (fun k => k ≤ k.minFac ^ n) |>.card
+
+/-- There exists m such that countExceeding n m > countNotExceeding n m. -/
+@[category undergraduate, AMS 11]
+theorem a_exists (n : ℕ) : ∃ m, countExceeding n m > countNotExceeding n m := by
+    sorry
+
+/-- The sequence a(n): least m such that countExceeding n m > countNotExceeding n m. -/
+noncomputable def a (n : ℕ) : ℕ :=
+  Nat.find (a_exists n)
+
+/-- a(1) = 15. -/
+@[category test, AMS 11]
+theorem a_one : a 1 = 15 := by
+  rw [a, Nat.find_eq_iff]
+  refine ⟨by decide, ?_⟩
+  intro m hm
+  interval_cases m <;> decide
+
+/-- a(2) = 27. -/
+@[category test, AMS 11]
+theorem a_two : a 2 = 27 := by
+  rw [a, Nat.find_eq_iff]
+  refine ⟨by decide, ?_⟩
+  intro m hm
+  interval_cases m <;> decide
+
+/-- a(3) = 57. -/
+@[category test, AMS 11]
+theorem a_three : a 3 = 57 := by
+  rw [a, Nat.find_eq_iff]
+  refine ⟨by decide, ?_⟩
+  intro m hm
+  interval_cases m <;> decide
+
+/-- The formula value. -/
+def formula (n : ℕ) : ℕ := 3 ^ n + 3 * 2 ^ n + 6
+
+/-- Conjecture: a(n) = 3^n + 3 * 2^n + 6 for n ≥ 1. -/
+@[category research formally solved using lean4 at "https://github.com/google-deepmind/formal-conjectures/pull/1894/commits/7a286754f623759d69a3dd18f482c53c1d70959b", AMS 11]
+theorem a_formula {n : ℕ} (hn : n ≥ 1) : a n = 3 ^ n + 3 * 2 ^ n + 6 := by
+  sorry
+
+end OeisA87719