slug: oeis-a140869-sundaram-triangle-complement-prime bibkey: librandi2012a140869 doi: null url: https://oeis.org/A140869 triage: theorem motivation_gids:

• D5/S3/Arith/Congruence/LibrandiSundaramTriangleComplementPrime

The A140869 Sundaram-type triangle complement

Problem

OEIS A140869, NAME (%N, verbatim):

T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1.

COMMENT (%C, verbatim; Vincenzo Librandi, Nov 18 2012):

Conjecture: If h does not belong to the sequence, then 4*h+5 is prime.

The literal claim is ∀ h, ¬(∃ m n, 1 ≤ n ≤ m ∧ T(m,n) = h) → 4h+5 prime. Here T(m,n) and the membership predicate use natural-number truncated subtraction and division. On the stated domain m ≥ n ≥ 1, the numerator 2mn+m+n−2 is at least 2, so these operations agree with the intended expression.

Motivation

The 2012 OEIS conjecture gives a universal complement-primality statement. The formal result proves that every natural value outside the triangle has prime 4h+5.

Gap

The dated surfaces recorded from preregistration issue #7527 and the probe were OEIS history revisions #1–#22, with revision #15 introducing the conjecture; no proof or refutation is recorded there. Sibling entries A153085 and A144652 were checked, and MathOverflow returned 0. arXiv and OpenAlex returned HTTP 429, and GitHub code search returned HTTP 401; these rate-limited surfaces are ASSUMED-UNVERIFIED. The statement is of Sundaram-sieve type, and no priority is claimed.

Route

Use the contrapositive. If 4h+5 is odd composite, choose odd factors d·e = 4h+5 with d,e ≥ 3, write d = 2a+1 and e = 2b+1, and order them so 1 ≤ n ≤ m after setting the smaller half-factor to n and the larger to m. Expanding gives 4ab+2a+2b+1 = 4h+5, hence a+b is even and the defining equation yields T(a,b) = h after the corresponding ordering.

Falsifier

An h outside the triangle for which 4*h+5 is composite would refute the theorem.

Evidence

• Lean module: D5/S3/Arith/Congruence/LibrandiSundaramTriangleComplementPrime.lean.
• Main theorem: librandi_a140869, with std3 axiom closure.
• The orchestrator checked h ≤ 3000: 2881 triangle values and 120 outside values, with zero counterexamples outside the triangle.
• The converse is false at h = 2, h = 8, and h = 12; this dossier does not claim it.

Triage

theorem

ASSUMED-UNVERIFIED

The arXiv/OpenAlex HTTP 429 and GitHub code-search HTTP 401 surfaces are rate-limited; the literature search is bounded, and no priority claim is made.