Conjectures associated with A067720

For members of the sequence other than 8, we have k + 1 is prime.

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-67720,
  title        = {Conjectures associated with A067720},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-67720}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

For members of the sequence other than $8$ , we have $k + 1$ is prime.

A067720 lists numbers $k$ such that $φ (k^{2} + 1) = k \cdot φ (k + 1)$ , where $φ$ is Euler's totient function.

The sequence exhibits a strong connection to primes: for almost all terms $k$ , $k + 1$ is prime. The conjecture states that $k = 8$ is the only exception.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«67720».

theorem prime_add_one_of_a {k : ℕ} (h : A k) (hne : k ≠ 8) : (k + 1).Prime

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

A067720

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.