Smallest composite c such that textrmprimorial(n) + c is prime

Conjecture: liminf_n → ∞ a(n)/p_n+1^2 = 1 < limsup_n → ∞ a(n)/p_n+1^2 = 2. - Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015

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-38771,
  title        = {Smallest composite c such that textrmprimorial(n) + c is prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-38771}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

conjecture1. Conjecture: $lim inf_{n \to \infty} \frac{a ( n )}{p _{n + 1}^{2}} = 1 <$ $lim sup_{n \to \infty} \frac{a ( n )}{p _{n + 1}^{2}} = 2$ .

Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015

conjecture2. All the terms in this sequence have exactly two prime factors. This conjecture is true for the first 133 terms.

Dmitry Kamenetsky, Jan 06 2019

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«38771» (2 statements).

theorem conjecture1 :
    let p_next_sq (n : ℕ) : ℝ := ((Nat.nth Nat.Prime n : ℝ)) ^ 2
    let seq (n : ℕ) : ℝ := (a n : ℝ) / p_next_sq n
    (liminf seq atTop = 1) ∧ (limsup seq atTop = 2)

theorem conjecture2 (n : ℕ) : (a n).IsSemiprime

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

A038771

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.