A113028 — solved through base 64

OEIS A113028: a(B) is the largest number whose base-B representation uses distinct nonzero digits and is divisible by every digit it contains — equivalently, divisible by the lcm of its digits.

The flavor of the problem in one value: a(16) = 0xFEDCB59726A1348 — fifteen distinct nonzero hex digits whose lcm, 360360, divides the value exactly. It is the one answer below you can check by eye, and it was first found in 2006, unsubmitted, while the sequence's author was advising this repository's author (Provenance). Today the engine returns it in 0.021 s — and a(64), the deliberate stopping point, is ~95 orders of magnitude larger.

This repository computes and proves a(B) for every base 2 ≤ B ≤ 64:

• Every value below carries a maximality proof — no conjectured or window-bounded entries remain. Labels: CERTIFIED (independent methods concordant) or STRONG (single-method exhaustive); definitions below.
• Bases 49–64 are first-ever computed values. The published OEIS b-file ends at base 48.
• The published a(46) is wrong — this project found a strictly larger arrangement of the same digit set. Corrected below.
• Bases 65–89 were swept and no completion was found — 22 bases refuted at their minimum widths, two bounded by budget, one structurally out of reach. Not a proof that none exist; see docs/results/HIGH-BASE-RANGE-65-89.md.
• Decimal expansions for all 63 terms: b113028.txt (OEIS b-file format, n = 2…64), machine-checked against this table by scripts/validate_bfile.py.

Digit values map to symbols (full spec, including the right-to-left handling of Hebrew digits, in docs/results/DIGIT-ALPHABET.md):

Most significant digit first; every Hebrew letter is followed by an invisible left-to-right mark so the string renders in place-value order.

• CERTIFIED — independent engine families or methods produced the same value with exhaustive coverage; a proof of maximality defended by concordance.
• STRONG — one engine family produced an exhaustive maximality argument (every prefix that could beat the value is refuted, exhausted, or bound-pruned). The residual risk is implementation error, not an unsearched region.
• verified — matches the published OEIS value; independently recomputed by this project's engines (bases 41–49 additionally re-certified by the autonomous bucket driver, and bases 2–39 corroborated exactly by an independent 2020 Ruby solver — see docs/results/PROVENANCE.md).

Per-base evidence, proof artifacts, and the full ladder definitions: docs/results/FRONTIER-STATUS.md. The complete maximality arguments for the hard bases (54, 59, 61, 62, 63, 64) — including a(64)'s one-line arithmetic obstruction — are in docs/results/MAXIMALITY-ARGUMENTS.md. Wall-clock records and the hardness metrics per base: docs/results/HARDNESS-AND-TIMINGS.md.

The published OEIS a(46) is suboptimal: it uses the correct (uniquely forced) digit set but drops digit 28 out of its descending slot. Verified independently twice, and re-certified by a second engine:

• published: 315044747190120671695735975284033252460559821155925276163089767538975200
• corrected: 315044747190120671695735975284412123404260147529994283460952247723479200

Details: docs/results/MAXIMALITY-ARGUMENTS.md.

Bases 65–89 (the exact ceiling of 128-bit arithmetic — lcm(1..89) overflows u128) were swept in a closed campaign: 22 bases refuted at their derived minimum widths, zero completions, b74/b82 bounded by budget, b86 out of the engine's structural reach. Every negative is bounded on both the width and release axes, so this is a statement about the campaign, not a proof about the range. Full record: docs/results/HIGH-BASE-RANGE-65-89.md. An external review of where the project should stop: docs/reviews/SHOULD-THE-PROJECT-STOP-AT-BASE-64.md.

The sequence began as Enigma 1343: Digital Dividend (New Scientist, 4 June 2005) and was submitted to OEIS by Peter Boothe in January 2006. Francis Carr extended it to a(13) within a month — and Jes Wolfe, working on the problem under Boothe's advice, reached base 16 that year without submitting it. The public trail then rested for sixteen years until Michael S. Branicky independently found a(14)–a(17) in 2022. Wolfe returned in 2024 with a(18)–a(21) and the published b-file through base 48, and in 2026 carried the sequence to base 64 with maximality proven rather than assumed — which makes the a(46) correction an erratum against our own earlier b-file, not a dispute with anyone. A 2020 Ruby solver by the same author independently corroborates bases 2–39 (38/38 exact). Full lineage and cross-check record: docs/results/PROVENANCE.md.

gcc -O2 -march=native -o a113028_v8 a113028_v8.c
./a113028_v8 2 48 1        # [lo] [hi] [verbose]

carrytrie.cpp is the production certification engine. Method overview: docs/engine/METHOD.md; development setup (the required git-hooks configuration) and engine history: docs/process/DEVELOPMENT.md.

After cloning, this is required, not optional:

git config core.hooksPath .githooks

CC0 1.0 Universal — public domain dedication. See LICENSE.