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Level A · Machine-checkable Hard Number theory P-oeis-114137

Difference between first odd semiprime > 2^n and 2^n

In this powers of 2 sequence, does 1 occur infinitely often?

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.

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Cite 
@misc{cairn-oeis-114137,
  title        = {Difference between first odd semiprime > 2^n and 2^n},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-114137}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

The question

conjecture1. In this powers of 2 sequence, does 1 occur infinitely often?

conjecture2. Does every odd number occur?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«114137» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem conjecture1 :
  answer(sorry) ↔ Set.Infinite {n : ℕ | a n = 1}
theorem conjecture2 :
  answer(sorry) ↔ ∀ k : ℕ, Odd k → ∃ n : ℕ, a n = k

What counts as progress

  • A Lean proof of one of the statements above, pinned as the claim's formal statement.
  • 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

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.