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import FormalConjecturesUtilPartitions of $n$ into distinct non-squarefree parts for $n > 23$
a: Number of partitions of $n$ into distinct parts that are not squarefree. This is the number of finite subsets of positive integers $P$ such that $\sum_{k \in P} k = n$ and every element $k \in P$ is not squarefree.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA256012open Nat Finseta: Number of partitions of $n$ into distinct parts that are not squarefree. This is the number of finite subsets of positive integers $P$ such that $\sum_{k \in P} k = n$ and every element $k \in P$ is not squarefree.
def a (n : ℕ) : ℕ :=
-- The parts must be $\le n$ to sum to $n$.
-- This is $\{1, 2, \dots, n\}$
let potential_parts : Finset ℕ := range (n + 1) \ {0}
-- We count all subsets P of potential_parts that satisfy the sum and the property.
card <| filter (fun P : Finset ℕ =>
P.sum id = n ∧
(∀ k ∈ P, ¬ Squarefree k)
) (powerset potential_parts)@[category test, AMS 11]
lemma a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_1 : a 1 = 0 := ⊢ a 1 = 0 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 0 := ⊢ a 2 = 0 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 0 := ⊢ a 3 = 0 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = 1 := ⊢ a 4 = 1 All goals completed! 🐙Conjecture: $a(n) > 0$ for $n > 23$.
A formal proof has been found with the methods described in arxiv/2605.22763.
@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/256012.wip.lean#L91"]
theorem a_pos_of_gt (n : ℕ) (hn : n > 23) : a n > 0 := n:ℕhn:n > 23⊢ a n > 0
All goals completed! 🐙end OeisA256012