Number of different products of subsets of 1, 2, …, n

Conjecture: let p ≤ 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

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-60957,
  title        = {Number of different products of subsets of 1, 2, …, n},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-60957}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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On the literature board

The problem

The question

Conjecture: let $p \leq n$ 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

The number of distinct products (including the empty product 1) of any subset of ${1, 2, \dots, n}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«60957».

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

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A060957

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.