bibkey: bouras2024a372763
authors: M. Bouras
year: 2024
title: "OEIS A372763: denominators of a continued fraction"
doi: null
url: https://oeis.org/A372763
claim: "Conjecture 2: Except for 2 and 3, all primes appear in the sequence once."
strata_touched:
  - D5/S3/Arith/Primes/BourasA372763OnceRefutation
license: citation-only
triage: anchor

Verified locator

Source: https://oeis.org/A372763

Bouras’s denominator sequence

OEIS A372763, NAME (unpaginated), states: “Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(…(n-1)-n/(n+5))))).”

Its COMMENTS (unpaginated) state: “Conjecture 1: The sequence contains only 1’s and the primes.” “Conjecture 2: Except for 2 and 3, all primes appear in the sequence once.”

The index range starts at three. Evaluate the inner continued fraction by the right fold of k − (k+1)/t over k = 2, …, n−1, with terminal rational value n+5; take its reciprocal and then the positive denominator in reduced form. The uniqueness assertion quantifies over every natural prime other than 2 and 3 and asks for exactly one index n ≥ 3.

The published data include a(10) = 11 and a(21) = 11. The refutation uses these two indices and makes no claim to discovering the numerical values. Both fractions have nonzero tails. It does not prove a global multiplicity classification or Conjecture 1.

Bouras, “A New Primes-Generating Sequence”, arXiv:2509.09745v1, https://arxiv.org/abs/2509.09745v1, Table 2 also displays the repeated value.

The numerical experiment is in the-omega-institute/trureturing-experiments, docs/reports/oeis-bouras-continued-fraction-occurrence-counts/, commit 4d3404c32ec26a2d3d54bf1ed150f97bc218a940. The check.py SHA-256 is 05b533313a609bd3bf139aef952a062566625488521351d2d2955731427d9ffb. The preregistration records python3 check.py 300 20000, exit 0, RESULT PASS. This finite computation is separate from the Lean proof of the repeated value.

Completeness of the external literature search and any assertion of publication priority are ASSUMED-UNVERIFIED. The source statements and experiment locator are recorded in https://github.com/the-omega-institute/trureturing/issues/18628.