1. FormalConjectures
  2. OEIS
  3. «87719»
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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:

namespace OeisA87719open 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 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.

⊢ countExceeding 1 15 > countNotExceeding 1 15 ∧ ∀ n < 15, ¬countExceeding 1 n > countNotExceeding 1 n refine ⟨⊢ countExceeding 1 15 > countNotExceeding 1 15 All goals completed! 🐙, ?_⟩ 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.

⊢ countExceeding 2 27 > countNotExceeding 2 27 ∧ ∀ n < 27, ¬countExceeding 2 n > countNotExceeding 2 n refine ⟨⊢ countExceeding 2 27 > countNotExceeding 2 27 All goals completed! 🐙, ?_⟩ 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.

⊢ countExceeding 3 57 > countNotExceeding 3 57 ∧ ∀ n < 57, ¬countExceeding 3 n > countNotExceeding 3 n refine ⟨⊢ countExceeding 3 57 > countNotExceeding 3 57 All goals completed! 🐙, ?_⟩ 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 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