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import FormalConjecturesUtil
namespace OeisA38107Number of primes strictly less than $n^2$.
def a (n : ℕ) : ℕ := (Nat.primesBelow (n ^ 2)).card
Value of the sequence a at 0.
@[category test, AMS 11]
theorem a_0 : a 0 = 0 := ⊢ a 0 = 0
All goals completed! 🐙
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 0 := ⊢ a 1 = 0
All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2
All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 4 := ⊢ a 3 = 4
All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 6 := ⊢ a 4 = 6
All goals completed! 🐙
Value of the sequence a at 5.
@[category test, AMS 11]
theorem a_5 : a 5 = 9 := ⊢ a 5 = 9
All goals completed! 🐙Conjecture: all the numbers $\sum_{i=j}^k \frac{1}{a(i)}$ with $1 < j \le k$ have pairwise distinct fractional parts.
Zhi-Wei Sun, Sep 24 2015
@[category research open, AMS 11]
theorem conjecture (j k j' k' : ℕ)
(hj : 1 < j) (hjk : j ≤ k) (hj' : 1 < j') (hj'k' : j' ≤ k')
(h_eq : Int.fract (∑ i ∈ Finset.Icc j k, (1 : ℝ) / (a i : ℝ)) =
Int.fract (∑ i ∈ Finset.Icc j' k', (1 : ℝ) / (a i : ℝ))) :
j = j' ∧ k = k' := j:ℕk:ℕj':ℕk':ℕhj:1 < jhjk:j ≤ khj':1 < j'hj'k':j' ≤ k'h_eq:Int.fract (∑ i ∈ Finset.Icc j k, 1 / ↑(a i)) = Int.fract (∑ i ∈ Finset.Icc j' k', 1 / ↑(a i))⊢ j = j' ∧ k = k'
All goals completed! 🐙end OeisA38107