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import FormalConjecturesUtil
import FormalConjectures.OEIS.«5153»Denominator of sum of reciprocals of divisors
Denominator of sum of reciprocals of divisors of $n$: $$\sum_{d \mid n} \frac{1}{d} = \frac{\sigma(n)}{n}$$ in lowest terms.
References:
namespace OeisA17666Denominator of sum of reciprocals of divisors of $n$.
def a (n : ℕ) : ℕ :=
if n = 0 then 1
else n / Nat.gcd n (∑ d ∈ n.divisors, d)@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 3 := ⊢ a 3 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 4 := ⊢ a 4 = 4 All goals completed! 🐙If $a(n)$ is in A005153, then $n$ is in A005153.
Jaycob Coleman, Sep 27 2014
We require $0 < n$ because $a(0) = 1$ is in A005153 (practical numbers), but $0$ is not.
@[category research open, AMS 11]
theorem conjecture (n : ℕ) (hn : 0 < n) : OeisA5153.A (a n) → OeisA5153.A n := n:ℕhn:0 < n⊢ OeisA5153.A (a n) → OeisA5153.A n
All goals completed! 🐙end OeisA17666