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import FormalConjecturesUtilNumber of different products of subsets of ${1, 2, \dots, n}$
The number of distinct products (including the empty product 1) of any subset of ${1, 2, \dots, n}$.
References:
namespace OeisA60957Number of different products of any subset of ${1, 2, \dots, n}$.
def a (n : ℕ) : ℕ :=
((Finset.Icc 1 n).powerset.image (·.prod id)).card@[category test, AMS 5 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1
All goals completed! 🐙@[category test, AMS 5 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1
All goals completed! 🐙@[category test, AMS 5 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2
All goals completed! 🐙@[category test, AMS 5 11]
theorem a_3 : a 3 = 4 := ⊢ a 3 = 4
All goals completed! 🐙@[category test, AMS 5 11]
theorem a_4 : a 4 = 8 := ⊢ a 4 = 8
All goals completed! 🐙@[category test, AMS 5 11]
theorem a_5 : a 5 = 16 := ⊢ a 5 = 16
All goals completed! 🐙The set of products of subsets of ${1, \dots, n}$.
def productsOfSubsets (n : ℕ) : Set ℕ :=
{m : ℕ | ∃ s ⊆ Finset.Icc 1 n, m = s.prod id}Conjecture: let $p \le n$ be prime. If $m$ and $p^a m$ are two such products, then so is $p^k m$ for all $0 < k < a$.
Yan Sheng Ang, Feb 13 2020
@[category research open, AMS 5 11]
theorem conjecture (n : ℕ) (p : ℕ) (hp : p.Prime) (hpn : p ≤ n)
(m a_exp : ℕ) (h1 : m ∈ productsOfSubsets n) (h2 : p ^ a_exp * m ∈ productsOfSubsets n)
(k : ℕ) (hk1 : 0 < k) (hk2 : k < a_exp) :
p ^ k * m ∈ productsOfSubsets n := n:ℕp:ℕhp:Nat.Prime phpn:p ≤ nm:ℕa_exp:ℕh1:m ∈ productsOfSubsets nh2:p ^ a_exp * m ∈ productsOfSubsets nk:ℕhk1:0 < khk2:k < a_exp⊢ p ^ k * m ∈ productsOfSubsets n
All goals completed! 🐙end OeisA60957