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import FormalConjecturesUtilProduct of composite numbers in triangular intervals
Product of all composite numbers between $n(n-1)/2+1$ and $n(n+1)/2$ (including boundaries), where $n(n-1)/2 = \binom{n}{2}$ and $n(n+1)/2 = \binom{n+1}{2}$.
References:
namespace OeisA93456Product of all composite numbers in the interval $[\binom{n}{2} + 1, \binom{n+1}{2}]$.
def a (n : ℕ) : ℕ :=
let L := n.choose 2 + 1
let R := (n + 1).choose 2
((Finset.Icc L R).filter fun k => 1 < k ∧ ¬ k.Prime).prod id
Value of the sequence a at 0.
@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 1 := ⊢ a 2 = 1 All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 24 := ⊢ a 3 = 24 All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 720 := ⊢ a 4 = 720 All goals completed! 🐙Conjecture: There are finitely many numbers such that $a(n)$ is not $\equiv 0 \pmod{a(n-1)}$. (Also mentioned in A093455.)
@[category research open, AMS 11]
theorem conjecture :
Set.Finite {n : ℕ | 1 < n ∧ ¬ (a (n - 1) ∣ a n)} := ⊢ {n | 1 < n ∧ ¬a (n - 1) ∣ a n}.Finite
All goals completed! 🐙end OeisA93456