1. FormalConjectures
  2. OEIS
  3. «194806»
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Size of smallest subset of ${1, 2, \dots, n}$ with distinct subset sums

Size of the smallest subset $S$ of $T = {1,2,3,\dots,n}$ such that $S \cdot S$ contains $T$, where $S \cdot S$ is the set of all products of elements of $S$.

References:

    A194806

    arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.

namespace OeisA194806open Finset Nat

The set of all products of elements from a Finset S.

def setProd (S : Finset ℕ) : Finset ℕ := (S.product S).image fun p : ℕ × ℕ => p.fst * p.snd

Size of the smallest subset $S$ of $T = {1,2,3,\dots,n}$ such that $S \cdot S$ contains $T$, where $S \cdot S$ is the set of all products of elements of $S$.

def a (n : ℕ) : ℕ := if h : n = 0 then 0 else let T_n := Icc 1 n -- The set of subsets $S \subseteq T_n$ such that $T_n \subseteq S \cdot S$. let valid_subsets : Finset (Finset ℕ) := T_n.powerset.filter (fun S : Finset ℕ => T_n ⊆ setProd S) -- Proof that $T_n$ is guaranteed to be a valid subset, ensuring `valid_subsets` is non-empty. have T_n_is_valid : T_n ∈ valid_subsets := n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}⊢ T_n ∈ valid_subsets n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}⊢ T_n ∈ T_n.powerset ∧ T_n ⊆ setProd T_n n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}⊢ T_n ∈ T_n.powersetn:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}⊢ T_n ⊆ setProd T_n -- 1. T_n ∈ T_n.powerset (i.e., T_n ⊆ T_n) n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}⊢ T_n ⊆ T_nn:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}⊢ T_n ⊆ setProd T_n; n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}⊢ T_n ⊆ setProd T_n -- 2. T_n ⊆ setProd T_n n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}k:ℕhk:k ∈ T_n⊢ k ∈ setProd T_n n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}k:ℕhk:k ∈ T_none_le_n:1 ≤ n⊢ k ∈ setProd T_n n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}k:ℕhk:k ∈ T_none_le_n:1 ≤ nh1:1 ∈ T_n⊢ k ∈ setProd T_n -- We show k = k * 1 is in setProd T_n -- setProd T_n is the image of T_n × T_n under multiplication. n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}k:ℕhk:k ∈ T_none_le_n:1 ≤ nh1:1 ∈ T_n⊢ ∃ a b, (a, b) ∈ T_n.product T_n ∧ a * b = k n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}k:ℕhk:k ∈ T_none_le_n:1 ≤ nh1:1 ∈ T_n⊢ (k, 1) ∈ T_n.product T_n ∧ k * 1 = k n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}k:ℕhk:k ∈ T_none_le_n:1 ≤ nh1:1 ∈ T_n⊢ (k, 1) ∈ T_n.product T_nn:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}k:ℕhk:k ∈ T_none_le_n:1 ≤ nh1:1 ∈ T_n⊢ k * 1 = k -- Show that (k, 1) ∈ T_n × T_n n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}k:ℕhk:k ∈ T_none_le_n:1 ≤ nh1:1 ∈ T_n⊢ (k, 1) ∈ T_n.product T_n All goals completed! 🐙 -- Show that k * 1 = k n:ℕh:¬n = 0T_n:Finset ℕ := Icc 1 nvalid_subsets:Finset (Finset ℕ) := {S ∈ T_n.powerset | T_n ⊆ setProd S}k:ℕhk:k ∈ T_none_le_n:1 ≤ nh1:1 ∈ T_n⊢ k * 1 = k All goals completed! 🐙 have h_nonempty : valid_subsets.Nonempty := ⟨T_n, T_n_is_valid⟩ let sizes := valid_subsets.image Finset.card -- The min' function requires proof that the finset is non-empty. have h_sizes_nonempty : sizes.Nonempty := h_nonempty.image Finset.card -- We return the minimum card of all valid subsets. sizes.min' h_sizes_nonempty@[category test, AMS 11] lemma a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11] lemma a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙@[category test, AMS 11] lemma a_3 : a 3 = 3 := ⊢ a 3 = 3 All goals completed! 🐙@[category test, AMS 11] lemma a_4 : a 4 = 3 := ⊢ a 4 = 3 All goals completed! 🐙@[category test, AMS 11] lemma a_5 : a 5 = 4 := ⊢ a 5 = 4 All goals completed! 🐙

Is $a(n) / \pi(n)$ bounded as $n \to \infty$? - Robert Israel, Jan 09 2017

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/194806.wip.lean#L472"] theorem a_div_prime_counting_bounded : ∃ C : ℝ, ∀ n : ℕ, 2 ≤ n → (a n : ℝ) / (Nat.primeCounting n : ℝ) ≤ C := ⊢ ∃ C, ∀ (n : ℕ), 2 ≤ n → ↑(OeisA194806.a n) / ↑n.primeCounting ≤ C All goals completed! 🐙end OeisA194806