/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
module
public 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:
@[expose] public sectionnamespace 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.) In fact there are infinitely many such numbers.
@[category research solved, AMS 11,
formal_proof using lean4 at "https://github.com/epoch-research/LeanOpenProblems-results/blob/f02efd9a8c5fc6a735d2a90c33e24f7278ce0ffc/runs/oeis-full-50usd-ant-j0j0g4uzligm1k41/oeis_93456_conjecture_0/Submission/Spec.lean#L109"]
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