OEIS A167857: three distinct prime divisors

The Online Encyclopedia of Integer Sequences, A167857, T. D. Noe, defines numbers whose divisors are represented by an integer polynomial:

That is, these numbers n have the property that there is a polynomial f(x) with integer coefficients whose values at x=0..tau(n)-1 are the divisors of n, where tau(n) is the number of divisors of n.

The published question is:

Are there terms with three distinct prime divisors?

The exact target is existence of a positive natural n with n.primeFactors.card = 3 and f : Polynomial Int such that, for every j < n.divisors.card, evaluation at the integer cast of j equals entry j of the actual increasingly sorted finset of all positive divisors of n. Both factor count and divisor list belong to this same n. The list has no duplicates; the inputs start at zero. No minimum, degree bound, numerical candidate, divisor subset, alternate order or rational-only conclusion is imposed. Preregistration: https://github.com/the-omega-institute/trureturing/issues/16601.

This is the original published existence question, classified as Tier 1 in the prospective registration. Its age is not evidence of novelty or priority.

Bounded source qualification inspected the current public/internal/history snapshots, seven direct neighbors, the Noe table, relevant repository source contracts, literature and official exact-target ownership queries. No whole-equivalent settlement or matching owner was found in those inspected scopes. Crossref/OpenAlex searches used top pages, some StackExchange pages remain unread, and GitHub divisor-polynomial search returned 100 metadata hits of 160. Unindexed/offline ownership remains unverified. This does not establish worldwide absence or priority.

Choose distinct primes p<q<r, each 1 modulo 5040, using the pinned Mathlib arbitrarily-large congruent-prime theorem. The same product n=pqr has eight positive divisors, all 1 modulo 5040. Let d_i be the actual increasing divisor list entry at i. The signed nodal denominator D_i = product_(j != i) (i-j) is nonzero and divides 5040 for each of the eight indices. Define c_i=(d_i-1)/D_i in Int and f=1+sum_i c_i*product_(j != i)(X-j) in Polynomial Int. Its rational image is the existing Lagrange interpolant plus 1, so the upstream node-evaluation theorem and injectivity of the cast prove every required integer evaluation.

A faithful refutation would negate the entire original existential, for all positive n with three distinct prime factors and every integer-coefficient polynomial. Failure of this route, one candidate or a finite search would not refute that existential.

The exact source-to-quantifier mapping is embodied in D5/S3/Arith/SortedDivisorPolynomialThreePrimes.result, with its mirrored Blueprint Scribe. The construction reuses pinned Mathlib; the proof shape is bind-only and the admission basis is open-problem-resolution. It is an unbounded symbolic existence proof, not a finite candidate certificate. The canonical frozen module contains seven included declarations. Complete-unit independent review, draft publication, the full original-BASE local gate, exact-head required CI, MERGED readback and subsequent dev verification remain outstanding; no counted delivery is claimed. The required information audit is unfinished: the compiled Reg source record is declared_unresolved with IE-C050 ClosedTruthReadout, rule=source.observation_data. Its missing source observation-data binding is tracked at https://github.com/the-omega-institute/trureturing/issues/16620.

What the settlement shows

Proved by D5/S3/Arith/SortedDivisorPolynomialThreePrimes.result: the decisive mechanism is a common residue modulus divisible by all eight Lagrange nodal denominators. This forces integral coefficients while allowing the actual sorted divisor order. Unproved in this delivery: a uniform extension to arbitrary prime-factor counts, minimal n and optimal degree. The latter two are outside the published question. No known-supplier, new-method or global novelty credit is claimed.

Frisch, Interpolation by Integer-Valued Polynomials, DOI10.1006/jabr.1998.7741: author/publisher fulltexts403. Chabert, About Polynomials Whose Divided Differences are Integer-Valued on Prime Numbers, DOI10.1080/00927872.2014.897588: publisher403, authoritative exact-title HAL query0; full theorem conditions unverified. The Volkov–Petrov paper arXiv:1108.3212v1 concerns quotient-field integer-valued interpolation; the fixed-divisor survey DOI10.5802/cml.54 concerns gcd/ideal conditions; the inspected scopes do not settle this whole ordered-list question. MathOverflow169083/169084 is conditional on integral existence. Math.SE4453859/1467910 gives general interpolation criteria; Math.SE4789468 uses divisor inputs and is a different problem.

Recent Zenodo DOI10.5281/zenodo.23148134 has an authoritative abstract about prime patterns, with fulltexts403. Immutable OEIS views403, two new arXiv429, prior arXiv export/timeouts/exact-ID/228/newest50/178-unread limits, Semantic Scholar429, two Crossref429, zbMATH403 and the AMS page870 mirror404 remain named access/scope gaps. BeautifulSoup/pdftotext were unavailable; pypdf missing-fontTools warnings limit extracted glyphs. Transport failures do not prove absence. The A073103 ID/description mismatch is excluded as a premise.