1. FormalConjectures
  2. OEIS
  3. «87719»
/- 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 FormalConjecturesUtil

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 $n \ge 1$.

Reference: A87719

namespace OeisA87719 open Nat

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 textbook, AMS 11] theorem declaration uses 'sorry'a_exists (n : ℕ) : ∃ m, countExceeding n m > countNotExceeding n m := n:ℕ⊢ ∃ m, countExceeding n m > countNotExceeding n m All goals completed! 🐙

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_1 : a 1 = 15 := ⊢ a 1 = 15 ⊢ countExceeding 1 15 > countNotExceeding 1 15 ∧ ∀ n < 15, ¬countExceeding 1 n > countNotExceeding 1 n refine ⟨⊢ countExceeding 1 15 > countNotExceeding 1 15 All goals completed! 🐙, ?_⟩ intro m m:ℕhm:m < 15⊢ ¬countExceeding 1 m > countNotExceeding 1 m m:ℕhm:0 < 15⊢ ¬countExceeding 1 0 > countNotExceeding 1 0m:ℕhm:1 < 15⊢ ¬countExceeding 1 1 > countNotExceeding 1 1m:ℕhm:2 < 15⊢ ¬countExceeding 1 2 > countNotExceeding 1 2m:ℕhm:3 < 15⊢ ¬countExceeding 1 3 > countNotExceeding 1 3m:ℕhm:4 < 15⊢ ¬countExceeding 1 4 > countNotExceeding 1 4m:ℕhm:5 < 15⊢ ¬countExceeding 1 5 > countNotExceeding 1 5m:ℕhm:6 < 15⊢ ¬countExceeding 1 6 > countNotExceeding 1 6m:ℕhm:7 < 15⊢ ¬countExceeding 1 7 > countNotExceeding 1 7m:ℕhm:8 < 15⊢ ¬countExceeding 1 8 > countNotExceeding 1 8m:ℕhm:9 < 15⊢ ¬countExceeding 1 9 > countNotExceeding 1 9m:ℕhm:10 < 15⊢ ¬countExceeding 1 10 > countNotExceeding 1 10m:ℕhm:11 < 15⊢ ¬countExceeding 1 11 > countNotExceeding 1 11m:ℕhm:12 < 15⊢ ¬countExceeding 1 12 > countNotExceeding 1 12m:ℕhm:13 < 15⊢ ¬countExceeding 1 13 > countNotExceeding 1 13m:ℕhm:14 < 15⊢ ¬countExceeding 1 14 > countNotExceeding 1 14 m:ℕhm:0 < 15⊢ ¬countExceeding 1 0 > countNotExceeding 1 0m:ℕhm:1 < 15⊢ ¬countExceeding 1 1 > countNotExceeding 1 1m:ℕhm:2 < 15⊢ ¬countExceeding 1 2 > countNotExceeding 1 2m:ℕhm:3 < 15⊢ ¬countExceeding 1 3 > countNotExceeding 1 3m:ℕhm:4 < 15⊢ ¬countExceeding 1 4 > countNotExceeding 1 4m:ℕhm:5 < 15⊢ ¬countExceeding 1 5 > countNotExceeding 1 5m:ℕhm:6 < 15⊢ ¬countExceeding 1 6 > countNotExceeding 1 6m:ℕhm:7 < 15⊢ ¬countExceeding 1 7 > countNotExceeding 1 7m:ℕhm:8 < 15⊢ ¬countExceeding 1 8 > countNotExceeding 1 8m:ℕhm:9 < 15⊢ ¬countExceeding 1 9 > countNotExceeding 1 9m:ℕhm:10 < 15⊢ ¬countExceeding 1 10 > countNotExceeding 1 10m:ℕhm:11 < 15⊢ ¬countExceeding 1 11 > countNotExceeding 1 11m:ℕhm:12 < 15⊢ ¬countExceeding 1 12 > countNotExceeding 1 12m:ℕhm:13 < 15⊢ ¬countExceeding 1 13 > countNotExceeding 1 13m:ℕhm:14 < 15⊢ ¬countExceeding 1 14 > countNotExceeding 1 14 All goals completed! 🐙

a(2) = 27.

@[category test, AMS 11] theorem a_2 : a 2 = 27 := ⊢ a 2 = 27 ⊢ countExceeding 2 27 > countNotExceeding 2 27 ∧ ∀ n < 27, ¬countExceeding 2 n > countNotExceeding 2 n refine ⟨⊢ countExceeding 2 27 > countNotExceeding 2 27 All goals completed! 🐙, ?_⟩ intro m m:ℕhm:m < 27⊢ ¬countExceeding 2 m > countNotExceeding 2 m m:ℕhm:0 < 27⊢ ¬countExceeding 2 0 > countNotExceeding 2 0m:ℕhm:1 < 27⊢ ¬countExceeding 2 1 > countNotExceeding 2 1m:ℕhm:2 < 27⊢ ¬countExceeding 2 2 > countNotExceeding 2 2m:ℕhm:3 < 27⊢ ¬countExceeding 2 3 > countNotExceeding 2 3m:ℕhm:4 < 27⊢ ¬countExceeding 2 4 > countNotExceeding 2 4m:ℕhm:5 < 27⊢ ¬countExceeding 2 5 > countNotExceeding 2 5m:ℕhm:6 < 27⊢ ¬countExceeding 2 6 > countNotExceeding 2 6m:ℕhm:7 < 27⊢ ¬countExceeding 2 7 > countNotExceeding 2 7m:ℕhm:8 < 27⊢ ¬countExceeding 2 8 > countNotExceeding 2 8m:ℕhm:9 < 27⊢ ¬countExceeding 2 9 > countNotExceeding 2 9m:ℕhm:10 < 27⊢ ¬countExceeding 2 10 > countNotExceeding 2 10m:ℕhm:11 < 27⊢ ¬countExceeding 2 11 > countNotExceeding 2 11m:ℕhm:12 < 27⊢ ¬countExceeding 2 12 > countNotExceeding 2 12m:ℕhm:13 < 27⊢ ¬countExceeding 2 13 > countNotExceeding 2 13m:ℕhm:14 < 27⊢ ¬countExceeding 2 14 > countNotExceeding 2 14m:ℕhm:15 < 27⊢ ¬countExceeding 2 15 > countNotExceeding 2 15m:ℕhm:16 < 27⊢ ¬countExceeding 2 16 > countNotExceeding 2 16m:ℕhm:17 < 27⊢ ¬countExceeding 2 17 > countNotExceeding 2 17m:ℕhm:18 < 27⊢ ¬countExceeding 2 18 > countNotExceeding 2 18m:ℕhm:19 < 27⊢ ¬countExceeding 2 19 > countNotExceeding 2 19m:ℕhm:20 < 27⊢ ¬countExceeding 2 20 > countNotExceeding 2 20m:ℕhm:21 < 27⊢ ¬countExceeding 2 21 > countNotExceeding 2 21m:ℕhm:22 < 27⊢ ¬countExceeding 2 22 > countNotExceeding 2 22m:ℕhm:23 < 27⊢ ¬countExceeding 2 23 > countNotExceeding 2 23m:ℕhm:24 < 27⊢ ¬countExceeding 2 24 > countNotExceeding 2 24m:ℕhm:25 < 27⊢ ¬countExceeding 2 25 > countNotExceeding 2 25m:ℕhm:26 < 27⊢ ¬countExceeding 2 26 > countNotExceeding 2 26 m:ℕhm:0 < 27⊢ ¬countExceeding 2 0 > countNotExceeding 2 0m:ℕhm:1 < 27⊢ ¬countExceeding 2 1 > countNotExceeding 2 1m:ℕhm:2 < 27⊢ ¬countExceeding 2 2 > countNotExceeding 2 2m:ℕhm:3 < 27⊢ ¬countExceeding 2 3 > countNotExceeding 2 3m:ℕhm:4 < 27⊢ ¬countExceeding 2 4 > countNotExceeding 2 4m:ℕhm:5 < 27⊢ ¬countExceeding 2 5 > countNotExceeding 2 5m:ℕhm:6 < 27⊢ ¬countExceeding 2 6 > countNotExceeding 2 6m:ℕhm:7 < 27⊢ ¬countExceeding 2 7 > countNotExceeding 2 7m:ℕhm:8 < 27⊢ ¬countExceeding 2 8 > countNotExceeding 2 8m:ℕhm:9 < 27⊢ ¬countExceeding 2 9 > countNotExceeding 2 9m:ℕhm:10 < 27⊢ ¬countExceeding 2 10 > countNotExceeding 2 10m:ℕhm:11 < 27⊢ ¬countExceeding 2 11 > countNotExceeding 2 11m:ℕhm:12 < 27⊢ ¬countExceeding 2 12 > countNotExceeding 2 12m:ℕhm:13 < 27⊢ ¬countExceeding 2 13 > countNotExceeding 2 13m:ℕhm:14 < 27⊢ ¬countExceeding 2 14 > countNotExceeding 2 14m:ℕhm:15 < 27⊢ ¬countExceeding 2 15 > countNotExceeding 2 15m:ℕhm:16 < 27⊢ ¬countExceeding 2 16 > countNotExceeding 2 16m:ℕhm:17 < 27⊢ ¬countExceeding 2 17 > countNotExceeding 2 17m:ℕhm:18 < 27⊢ ¬countExceeding 2 18 > countNotExceeding 2 18m:ℕhm:19 < 27⊢ ¬countExceeding 2 19 > countNotExceeding 2 19m:ℕhm:20 < 27⊢ ¬countExceeding 2 20 > countNotExceeding 2 20m:ℕhm:21 < 27⊢ ¬countExceeding 2 21 > countNotExceeding 2 21m:ℕhm:22 < 27⊢ ¬countExceeding 2 22 > countNotExceeding 2 22m:ℕhm:23 < 27⊢ ¬countExceeding 2 23 > countNotExceeding 2 23m:ℕhm:24 < 27⊢ ¬countExceeding 2 24 > countNotExceeding 2 24m:ℕhm:25 < 27⊢ ¬countExceeding 2 25 > countNotExceeding 2 25m:ℕhm:26 < 27⊢ ¬countExceeding 2 26 > countNotExceeding 2 26 All goals completed! 🐙

a(3) = 57.

@[category test, AMS 11] theorem a_3 : a 3 = 57 := ⊢ a 3 = 57 ⊢ countExceeding 3 57 > countNotExceeding 3 57 ∧ ∀ n < 57, ¬countExceeding 3 n > countNotExceeding 3 n refine ⟨⊢ countExceeding 3 57 > countNotExceeding 3 57 All goals completed! 🐙, ?_⟩ intro m m:ℕhm:m < 57⊢ ¬countExceeding 3 m > countNotExceeding 3 m m:ℕhm:0 < 57⊢ ¬countExceeding 3 0 > countNotExceeding 3 0m:ℕhm:1 < 57⊢ ¬countExceeding 3 1 > countNotExceeding 3 1m:ℕhm:2 < 57⊢ ¬countExceeding 3 2 > countNotExceeding 3 2m:ℕhm:3 < 57⊢ ¬countExceeding 3 3 > countNotExceeding 3 3m:ℕhm:4 < 57⊢ ¬countExceeding 3 4 > countNotExceeding 3 4m:ℕhm:5 < 57⊢ ¬countExceeding 3 5 > countNotExceeding 3 5m:ℕhm:6 < 57⊢ ¬countExceeding 3 6 > countNotExceeding 3 6m:ℕhm:7 < 57⊢ ¬countExceeding 3 7 > countNotExceeding 3 7m:ℕhm:8 < 57⊢ ¬countExceeding 3 8 > countNotExceeding 3 8m:ℕhm:9 < 57⊢ ¬countExceeding 3 9 > countNotExceeding 3 9m:ℕhm:10 < 57⊢ ¬countExceeding 3 10 > countNotExceeding 3 10m:ℕhm:11 < 57⊢ ¬countExceeding 3 11 > countNotExceeding 3 11m:ℕhm:12 < 57⊢ ¬countExceeding 3 12 > countNotExceeding 3 12m:ℕhm:13 < 57⊢ ¬countExceeding 3 13 > countNotExceeding 3 13m:ℕhm:14 < 57⊢ ¬countExceeding 3 14 > countNotExceeding 3 14m:ℕhm:15 < 57⊢ ¬countExceeding 3 15 > countNotExceeding 3 15m:ℕhm:16 < 57⊢ ¬countExceeding 3 16 > countNotExceeding 3 16m:ℕhm:17 < 57⊢ ¬countExceeding 3 17 > countNotExceeding 3 17m:ℕhm:18 < 57⊢ ¬countExceeding 3 18 > countNotExceeding 3 18m:ℕhm:19 < 57⊢ ¬countExceeding 3 19 > countNotExceeding 3 19m:ℕhm:20 < 57⊢ ¬countExceeding 3 20 > countNotExceeding 3 20m:ℕhm:21 < 57⊢ ¬countExceeding 3 21 > countNotExceeding 3 21m:ℕhm:22 < 57⊢ ¬countExceeding 3 22 > countNotExceeding 3 22m:ℕhm:23 < 57⊢ ¬countExceeding 3 23 > countNotExceeding 3 23m:ℕhm:24 < 57⊢ ¬countExceeding 3 24 > countNotExceeding 3 24m:ℕhm:25 < 57⊢ ¬countExceeding 3 25 > countNotExceeding 3 25m:ℕhm:26 < 57⊢ ¬countExceeding 3 26 > countNotExceeding 3 26m:ℕhm:27 < 57⊢ ¬countExceeding 3 27 > countNotExceeding 3 27m:ℕhm:28 < 57⊢ ¬countExceeding 3 28 > countNotExceeding 3 28m:ℕhm:29 < 57⊢ ¬countExceeding 3 29 > countNotExceeding 3 29m:ℕhm:30 < 57⊢ ¬countExceeding 3 30 > countNotExceeding 3 30m:ℕhm:31 < 57⊢ ¬countExceeding 3 31 > countNotExceeding 3 31m:ℕhm:32 < 57⊢ ¬countExceeding 3 32 > countNotExceeding 3 32m:ℕhm:33 < 57⊢ ¬countExceeding 3 33 > countNotExceeding 3 33m:ℕhm:34 < 57⊢ ¬countExceeding 3 34 > countNotExceeding 3 34m:ℕhm:35 < 57⊢ ¬countExceeding 3 35 > countNotExceeding 3 35m:ℕhm:36 < 57⊢ ¬countExceeding 3 36 > countNotExceeding 3 36m:ℕhm:37 < 57⊢ ¬countExceeding 3 37 > countNotExceeding 3 37m:ℕhm:38 < 57⊢ ¬countExceeding 3 38 > countNotExceeding 3 38m:ℕhm:39 < 57⊢ ¬countExceeding 3 39 > countNotExceeding 3 39m:ℕhm:40 < 57⊢ ¬countExceeding 3 40 > countNotExceeding 3 40m:ℕhm:41 < 57⊢ ¬countExceeding 3 41 > countNotExceeding 3 41m:ℕhm:42 < 57⊢ ¬countExceeding 3 42 > countNotExceeding 3 42m:ℕhm:43 < 57⊢ ¬countExceeding 3 43 > countNotExceeding 3 43m:ℕhm:44 < 57⊢ ¬countExceeding 3 44 > countNotExceeding 3 44m:ℕhm:45 < 57⊢ ¬countExceeding 3 45 > countNotExceeding 3 45m:ℕhm:46 < 57⊢ ¬countExceeding 3 46 > countNotExceeding 3 46m:ℕhm:47 < 57⊢ ¬countExceeding 3 47 > countNotExceeding 3 47m:ℕhm:48 < 57⊢ ¬countExceeding 3 48 > countNotExceeding 3 48m:ℕhm:49 < 57⊢ ¬countExceeding 3 49 > countNotExceeding 3 49m:ℕhm:50 < 57⊢ ¬countExceeding 3 50 > countNotExceeding 3 50m:ℕhm:51 < 57⊢ ¬countExceeding 3 51 > countNotExceeding 3 51m:ℕhm:52 < 57⊢ ¬countExceeding 3 52 > countNotExceeding 3 52m:ℕhm:53 < 57⊢ ¬countExceeding 3 53 > countNotExceeding 3 53m:ℕhm:54 < 57⊢ ¬countExceeding 3 54 > countNotExceeding 3 54m:ℕhm:55 < 57⊢ ¬countExceeding 3 55 > countNotExceeding 3 55m:ℕhm:56 < 57⊢ ¬countExceeding 3 56 > countNotExceeding 3 56 m:ℕhm:0 < 57⊢ ¬countExceeding 3 0 > countNotExceeding 3 0m:ℕhm:1 < 57⊢ ¬countExceeding 3 1 > countNotExceeding 3 1m:ℕhm:2 < 57⊢ ¬countExceeding 3 2 > countNotExceeding 3 2m:ℕhm:3 < 57⊢ ¬countExceeding 3 3 > countNotExceeding 3 3m:ℕhm:4 < 57⊢ ¬countExceeding 3 4 > countNotExceeding 3 4m:ℕhm:5 < 57⊢ ¬countExceeding 3 5 > countNotExceeding 3 5m:ℕhm:6 < 57⊢ ¬countExceeding 3 6 > countNotExceeding 3 6m:ℕhm:7 < 57⊢ ¬countExceeding 3 7 > countNotExceeding 3 7m:ℕhm:8 < 57⊢ ¬countExceeding 3 8 > countNotExceeding 3 8m:ℕhm:9 < 57⊢ ¬countExceeding 3 9 > countNotExceeding 3 9m:ℕhm:10 < 57⊢ ¬countExceeding 3 10 > countNotExceeding 3 10m:ℕhm:11 < 57⊢ ¬countExceeding 3 11 > countNotExceeding 3 11m:ℕhm:12 < 57⊢ ¬countExceeding 3 12 > countNotExceeding 3 12m:ℕhm:13 < 57⊢ ¬countExceeding 3 13 > countNotExceeding 3 13m:ℕhm:14 < 57⊢ ¬countExceeding 3 14 > countNotExceeding 3 14m:ℕhm:15 < 57⊢ ¬countExceeding 3 15 > countNotExceeding 3 15m:ℕhm:16 < 57⊢ ¬countExceeding 3 16 > countNotExceeding 3 16m:ℕhm:17 < 57⊢ ¬countExceeding 3 17 > countNotExceeding 3 17m:ℕhm:18 < 57⊢ ¬countExceeding 3 18 > countNotExceeding 3 18m:ℕhm:19 < 57⊢ ¬countExceeding 3 19 > countNotExceeding 3 19m:ℕhm:20 < 57⊢ ¬countExceeding 3 20 > countNotExceeding 3 20m:ℕhm:21 < 57⊢ ¬countExceeding 3 21 > countNotExceeding 3 21m:ℕhm:22 < 57⊢ ¬countExceeding 3 22 > countNotExceeding 3 22m:ℕhm:23 < 57⊢ ¬countExceeding 3 23 > countNotExceeding 3 23m:ℕhm:24 < 57⊢ ¬countExceeding 3 24 > countNotExceeding 3 24m:ℕhm:25 < 57⊢ ¬countExceeding 3 25 > countNotExceeding 3 25m:ℕhm:26 < 57⊢ ¬countExceeding 3 26 > countNotExceeding 3 26m:ℕhm:27 < 57⊢ ¬countExceeding 3 27 > countNotExceeding 3 27m:ℕhm:28 < 57⊢ ¬countExceeding 3 28 > countNotExceeding 3 28m:ℕhm:29 < 57⊢ ¬countExceeding 3 29 > countNotExceeding 3 29m:ℕhm:30 < 57⊢ ¬countExceeding 3 30 > countNotExceeding 3 30m:ℕhm:31 < 57⊢ ¬countExceeding 3 31 > countNotExceeding 3 31m:ℕhm:32 < 57⊢ ¬countExceeding 3 32 > countNotExceeding 3 32m:ℕhm:33 < 57⊢ ¬countExceeding 3 33 > countNotExceeding 3 33m:ℕhm:34 < 57⊢ ¬countExceeding 3 34 > countNotExceeding 3 34m:ℕhm:35 < 57⊢ ¬countExceeding 3 35 > countNotExceeding 3 35m:ℕhm:36 < 57⊢ ¬countExceeding 3 36 > countNotExceeding 3 36m:ℕhm:37 < 57⊢ ¬countExceeding 3 37 > countNotExceeding 3 37m:ℕhm:38 < 57⊢ ¬countExceeding 3 38 > countNotExceeding 3 38m:ℕhm:39 < 57⊢ ¬countExceeding 3 39 > countNotExceeding 3 39m:ℕhm:40 < 57⊢ ¬countExceeding 3 40 > countNotExceeding 3 40m:ℕhm:41 < 57⊢ ¬countExceeding 3 41 > countNotExceeding 3 41m:ℕhm:42 < 57⊢ ¬countExceeding 3 42 > countNotExceeding 3 42m:ℕhm:43 < 57⊢ ¬countExceeding 3 43 > countNotExceeding 3 43m:ℕhm:44 < 57⊢ ¬countExceeding 3 44 > countNotExceeding 3 44m:ℕhm:45 < 57⊢ ¬countExceeding 3 45 > countNotExceeding 3 45m:ℕhm:46 < 57⊢ ¬countExceeding 3 46 > countNotExceeding 3 46m:ℕhm:47 < 57⊢ ¬countExceeding 3 47 > countNotExceeding 3 47m:ℕhm:48 < 57⊢ ¬countExceeding 3 48 > countNotExceeding 3 48m:ℕhm:49 < 57⊢ ¬countExceeding 3 49 > countNotExceeding 3 49m:ℕhm:50 < 57⊢ ¬countExceeding 3 50 > countNotExceeding 3 50m:ℕhm:51 < 57⊢ ¬countExceeding 3 51 > countNotExceeding 3 51m:ℕhm:52 < 57⊢ ¬countExceeding 3 52 > countNotExceeding 3 52m:ℕhm:53 < 57⊢ ¬countExceeding 3 53 > countNotExceeding 3 53m:ℕhm:54 < 57⊢ ¬countExceeding 3 54 > countNotExceeding 3 54m:ℕhm:55 < 57⊢ ¬countExceeding 3 55 > countNotExceeding 3 55m:ℕhm:56 < 57⊢ ¬countExceeding 3 56 > countNotExceeding 3 56 All goals completed! 🐙

We have the following formula: $a(n) = 3^n + 3 * 2^n + 6$ for $n \geq 1$.

@[category textbook, AMS 11, formal_proof using formal_conjectures at "https://github.com/google-deepmind/formal-conjectures/pull/1894/commits/7a286754f623759d69a3dd18f482c53c1d70959b"] theorem declaration uses 'sorry'a_formula {n : ℕ} (hn : n ≥ 1) : a n = 3 ^ n + 3 * 2 ^ n + 6 := n:ℕhn:n ≥ 1⊢ a n = 3 ^ n + 3 * 2 ^ n + 6 All goals completed! 🐙 end OeisA87719