bibkey: hasler2023a359482 authors: Eric Angelini, Hans Havermann, and M. F. Hasler year: 2023 title: OEIS A359482 and the single-digit conjecture doi: null url: https://oeis.org/A359482 claim: “A359482 is not a permutation of the positive integers; its only single-digit term is its initial 1.” strata_touched:

• D5/S3/Arith/SumInConcatenation license: citation-only triage: anchor

OEIS A359482

The entry defines the lexicographically earliest sequence of distinct positive terms whose adjacent sum is a substring of their decimal concatenation. It asks whether this is a permutation. Hasler’s July 3, 2023 comment conjectures that it is not, and that the initial 1 is its only single-digit term. The same comment makes a stronger assertion about multiples of ten below 100; the present result does not address that assertion.

Verified locator

doi: null url: https://oeis.org/A359482

The sources are https://oeis.org/A359482/internal and section F of Eric Angelini’s July 2023 blog, including the Hans/Hasler corrections: https://cinquantesignes.blogspot.com/2023/07/echecs-et-maths.html The incorrect prefix with 899 is corrected to 889. These sources give the question, conjecture, prefixes and programs, but no proof of single-digit exclusion.

Proof scope

Mathlib supplies digit length monotonicity, base evaluation, concatenation, and list infix lemmas. The proof of the target theorem is a repository derivation; no publication priority is claimed.

Mathematical scope

The repository proof handles all positive x and all 1<=d<=9. Reversing the decimal lists leaves either the entire old operand or a shifted list. In the shifted case, evaluation forces a10^length(r)=9value(r); reduction modulo nine gives a=0 or a=9, and each contradicts positivity or the remainder bound. This includes the empty remainder list, so one-digit x is covered.

Greedy totality is proved separately: repeating the decimal block of x and then appending a block of zeros yields arbitrarily large legal successors. Thus a positive unused successor always exists. The sequence is defined by the original minimum rule, and the missing positive integer 2 proves failure of surjectivity.

Unverified scope

The OEIS revision history, Havermann’s linked 100000-term file, and the one-hop A300000 entry were not opened by this implementation seat (ASSUMED-UNVERIFIED). No exhaustive literature-absence claim is made.