FormalConjectures.OEIS.«7468»

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import FormalConjecturesUtil

Sum of next $n$ primes

The sum of the primes in the $n$-th row of the prime number triangle: $$a(n) = \sum_{i = 1 + n(n-1)/2}^{n + n(n-1)/2} p_i$$ with $a(0) = 0$.

References:

A007468

namespace OeisA7468

Sum of the next $n$ primes, with $a(0) = 0$.

noncomputable def a (n : ℕ) : ℕ :=
  let startIdx : ℕ := (n * (n - 1)) / 2
  ∑ i ∈ Finset.range n, Nat.nth Nat.Prime (startIdx + i)

Value of the sequence a at 0.

@[category test, AMS 11]
theorem a_0 : a 0 = 0 := by⊢ a 0 = 0 rflAll goals completed! 🐙

Value of the sequence a at 1.


@[category test, AMS 11]
theorem a_1 : a 1 = 2 := by⊢ a 1 = 2
  dsimp [a]⊢ ∑ i ∈ {0}, Nat.nth Nat.Prime (0 + i) = 2
  rw [Finset.sum_singleton,⊢ Nat.nth Nat.Prime (0 + 0) = 2All goals completed! 🐙 add_zero,⊢ Nat.nth Nat.Prime 0 = 2All goals completed! 🐙 Nat.nth_prime_zero_eq_two⊢ 2 = 2All goals completed! 🐙]All goals completed! 🐙

Value of the sequence a at 2.


@[category test, AMS 11]
theorem a_2 : a 2 = 8 := by⊢ a 2 = 8
  dsimp [a]⊢ ∑ i ∈ Finset.range 2, Nat.nth Nat.Prime (1 + i) = 8
  rw [Finset.sum_range_succ,⊢ ∑ x ∈ Finset.range 1, Nat.nth Nat.Prime (1 + x) + Nat.nth Nat.Prime (1 + 1) = 8All goals completed! 🐙 Finset.sum_range_one,⊢ Nat.nth Nat.Prime (1 + 0) + Nat.nth Nat.Prime (1 + 1) = 8All goals completed! 🐙 add_zero,⊢ Nat.nth Nat.Prime 1 + Nat.nth Nat.Prime (1 + 1) = 8All goals completed! 🐙 Nat.nth_prime_one_eq_three,⊢ 3 + Nat.nth Nat.Prime (1 + 1) = 8All goals completed! 🐙
    show (1 + 1 : ℕ) = 2 by⊢ a 2 = 8All goals completed! 🐙 rflAll goals completed! 🐙All goals completed! 🐙, Nat.nth_prime_two_eq_five⊢ 3 + 5 = 8All goals completed! 🐙]All goals completed! 🐙

Value of the sequence a at 3.


@[category test, AMS 11]
theorem a_3 : a 3 = 31 := by⊢ a 3 = 31
  dsimp [a]⊢ ∑ i ∈ Finset.range 3, Nat.nth Nat.Prime (3 + i) = 31
  have h5 : Nat.nth Nat.Prime 5 = 13 := Nat.nth_count (by⊢ Nat.Prime 13h5:Nat.nth Nat.Prime 5 = 13⊢ ∑ i ∈ Finset.range 3, Nat.nth Nat.Prime (3 + i) = 31 decideAll goals completed! 🐙h5:Nat.nth Nat.Prime 5 = 13⊢ ∑ i ∈ Finset.range 3, Nat.nth Nat.Prime (3 + i) = 31 : Nat.Prime 13)h5:Nat.nth Nat.Prime 5 = 13⊢ ∑ i ∈ Finset.range 3, Nat.nth Nat.Prime (3 + i) = 31
  rw [Finset.sum_range_succ,h5:Nat.nth Nat.Prime 5 = 13⊢ ∑ x ∈ Finset.range 2, Nat.nth Nat.Prime (3 + x) + Nat.nth Nat.Prime (3 + 2) = 31All goals completed! 🐙 Finset.sum_range_succ,h5:Nat.nth Nat.Prime 5 = 13⊢ ∑ x ∈ Finset.range 1, Nat.nth Nat.Prime (3 + x) + Nat.nth Nat.Prime (3 + 1) + Nat.nth Nat.Prime (3 + 2) = 31All goals completed! 🐙 Finset.sum_range_one,h5:Nat.nth Nat.Prime 5 = 13⊢ Nat.nth Nat.Prime (3 + 0) + Nat.nth Nat.Prime (3 + 1) + Nat.nth Nat.Prime (3 + 2) = 31All goals completed! 🐙 add_zero,h5:Nat.nth Nat.Prime 5 = 13⊢ Nat.nth Nat.Prime 3 + Nat.nth Nat.Prime (3 + 1) + Nat.nth Nat.Prime (3 + 2) = 31All goals completed! 🐙
    Nat.nth_prime_three_eq_seven,h5:Nat.nth Nat.Prime 5 = 13⊢ 7 + Nat.nth Nat.Prime (3 + 1) + Nat.nth Nat.Prime (3 + 2) = 31All goals completed! 🐙 show (3 + 1 : ℕ) = 4 by⊢ a 3 = 31All goals completed! 🐙 rflAll goals completed! 🐙All goals completed! 🐙, Nat.nth_prime_four_eq_eleven,h5:Nat.nth Nat.Prime 5 = 13⊢ 7 + 11 + Nat.nth Nat.Prime (3 + 2) = 31All goals completed! 🐙
    show (3 + 2 : ℕ) = 5 by⊢ a 3 = 31All goals completed! 🐙 rflAll goals completed! 🐙All goals completed! 🐙, h5h5:Nat.nth Nat.Prime 5 = 13⊢ 7 + 11 + 13 = 31All goals completed! 🐙]All goals completed! 🐙

The only positive integer $n$ such that $a(n)$ is a perfect square is $n=38$.

Carlos Eduardo Olivieri, Mar 09 2015

@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 0 < n) (hsq : IsSquare (a n)) : n = 38 := byn:ℕhn:0 < nhsq:IsSquare (a n)⊢ n = 38
  sorryAll goals completed! 🐙

end OeisA7468