slug: oeis-a110454-distinct-partition-concatenation-zero-indices bibkey: murthy2005a110454 doi: null url: https://oeis.org/A110454 triage: theorem motivation_gids:

• D5/S1/Digit/Admissibility/DistinctPartitionConcatenationComposite

The zero indices of A110454

Problem

OEIS A110454, NAME (verbatim):

Largest composite number obtained by concatenation of parts of a distinct partition of n, or 0 if no such number exist.

COMMENT (verbatim; Amarnath Murthy, Aug 04 2005):

Conjecture a(n) = 0 only for n = 1, 2 and 4.

The Maple program by R. J. Mathar, dated Feb 08 2008, examines every permutation of every distinct partition and accepts a concatenation exactly when m > 4 and m is not prime. Its loop includes single-part partitions.

Motivation

The conjecture identifies every positive index where the admissible set is empty. A proof therefore requires both an unbounded construction away from the three exceptional indices and finite exclusion at 1, 2, and 4.

Gap

The dated search recorded on 2026-09-13 checked the OEIS entry and all 11 revisions, arXiv by identifier, OpenAlex full text, MathOverflow advanced search, GitHub issues, commits, and repositories outside this repository, Loogle, and LeanSearch. A proof or counterexample was not found in the checked surfaces. GitHub code-content search, Google Scholar, and MathSciNet were not checked. This bounded search is not an exhaustive literature or priority claim.

Route

At n = 3, the ordered distinct partition [2,1] gives the concatenation 21, which is greater than four and composite. For every n >= 5, the ordered distinct partition [n-2,2] gives 10(n-2)+2 = 2(5n-9), again greater than four and composite. The remaining positive indices are 1, 2, and 4; direct classification of their distinct partitions shows that none of their concatenations is both greater than four and composite.

Falsifier

A positive n outside {1,2,4} for which every distinct-partition concatenation greater than four is prime would contradict the first conjunct. An admissible composite concatenation at any of 1, 2, or 4 would contradict the second conjunct.

Evidence

• Lean module: D5/S1/Digit/Admissibility/DistinctPartitionConcatenationComposite.lean.
• Main theorem: murthy_conjecture.
• Public definitions: decimalConcatenation and C.
• The theorem’s axiom closure is exactly propext, Classical.choice, and Quot.sound.
• The exact check recorded by the orchestrator enumerated all distinct partitions and their permutations for 1 <= n <= 35; its values matched the OEIS DATA at every index, and its zero indices were exactly 1, 2, and 4.

Triage

theorem. The formal proof establishes both the universal existence clause outside the three exceptions and the complete finite exclusion clause at the exceptions.

ASSUMED-UNVERIFIED

The revision-history and literature-search report is evidence recorded in the preregistration issue and was not independently repeated here. The enumeration through n = 35 is an orchestrator-supplied exact check, not a kernel proof. Source-to-Lean identification is not itself kernel checked. No exhaustive literature or first-publication priority claim follows.