1. FormalConjectures
  2. OEIS
  3. «64169»
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Numerator - denominator in $n$-th harmonic number

$a(n) = \text{numerator}(H_n) - \text{denominator}(H_n)$, where $H_n = 1 + 1/2 + \dots + 1/n$.

References:

namespace OeisA64169

The $n$-th harmonic number $H_n = \sum_{k=1}^n \frac{1}{k}$ as a rational number.

def H (n : ℕ) : ℚ := ∑ k ∈ Finset.Icc 1 n, 1 / (k : ℚ)

$a(n) = \text{numerator}(H_n) - \text{denominator}(H_n)$.

def a (n : ℕ) : ℤ := (H n).num - (H n).den@[category test, AMS 11] theorem a_1 : a 1 = 0 := ⊢ a 1 = 0 All goals completed! 🐙@[category test, AMS 11] theorem a_2 : a 2 = 1 := ⊢ a 2 = 1 All goals completed! 🐙@[category test, AMS 11] theorem a_3 : a 3 = 5 := ⊢ a 3 = 5 All goals completed! 🐙@[category test, AMS 11] theorem a_4 : a 4 = 13 := ⊢ a 4 = 13 All goals completed! 🐙@[category test, AMS 11] theorem a_5 : a 5 = 77 := ⊢ a 5 = 77 All goals completed! 🐙

"Conjecture: for $n > 2$, $n$ divides $a(n-2)$ if and only if $n$ is a prime. Checked up to 20000. - Amiram Eldar and Thomas Ordowski, Jul 27 2019"

@[category research open, AMS 11] theorem conjecture (n : ℕ) (hn : 2 < n) : (n : ℤ) ∣ a (n - 2) ↔ n.Prime := n:ℕhn:2 < n⊢ ↑n ∣ a (n - 2) ↔ Nat.Prime n All goals completed! 🐙end OeisA64169