# P2658: Longest rainbow divisor-count interval below 10^12 - ID: `P2658` - Reference: `rainbow-divisor-count-run-1e12` - Page: https://theoremdb.org/statements/P2658 - Record maturity: Reviewed problem with recorded work ## Problem Let \(\tau(n)\) denote the number of positive divisors of \(n\). Determine the largest \(L\) for which some \(m\) satisfies \(m+L-1\le10^{12}\) and the values \(\tau(m),\tau(m+1),\ldots,\tau(m+L-1)\) are pairwise distinct. ### Context This finite search asks for the longest consecutive interval below the cutoff whose integers all have different divisor counts. ### Definitions - **Definition (divisor function).** The divisor function \(\tau(n)\) is the number of positive divisors of \(n\). - **Definition (rainbow interval).** A rainbow interval is a consecutive interval on which the divisor-count values are pairwise distinct. ### What counts as a solution - Give a maximum interval with certified factorizations of all its terms, plus a complete segmented sweep proving that no longer rainbow interval ends below 10^12. ## Status A length-14 rainbow divisor-count interval is certified, and exhaustive search proves 14 is the exact maximum through endpoint \(10^{10}\); whether a longer interval ends between \(10^{10}\) and \(10^{12}\) remains open. [4](#reference-4) ## Work ### Evidence for the current status **Computation 1 (The requested maximum is at least fourteen).** A length-14 rainbow divisor-count interval is certified, and exhaustive search proves 14 is the exact maximum through endpoint \(10^{10}\); whether a longer interval ends between \(10^{10}\) and \(10^{12}\) remains open. The interval \[ 1745175039\leq n\leq1745175052 \] has divisor-count vector \[ (12,80,16,96,8,6,32,4,24,128,2,48,20,36). \] All fourteen entries are different. Its final integer is below \(10^{12}\), so the candidate's maximum \(L\) satisfies \[ L\geq14. \] The companion sweep finds this as the first length-14 record. The global upper bound requested by the candidate remains open in this fixture because the exhaustive computation stops at \(10^{10}\). ### Background and intake notes - Original intake status: Status remains unverified. Runs of equal divisor counts are tabulated, while the pairwise-distinct variant appears to have much less standard record data. - Use a segmented factor sieve and a sliding map from divisor counts to their most recent positions. Carry both the current window and its last occurrences across segment boundaries. - Trap: a segment-local maximum can miss an interval crossing a boundary. Probable-prime cofactors also make divisor counts conditional. - Recorded example: The interval 590890 through 590900 has divisor counts 16,12,24,6,8,20,10,4,32,2,36, all distinct. ### Other known results - **Theorem 1** (established): Trial division through the square root gives complete prime factorizations and reproduces all fourteen divisor counts. [4](#reference-4) - **Computation 2** (reproduced): A complete sweep of all ten billion divisor counts proves that no length-15 rainbow interval ends by 10,000,000,000. [4](#reference-4) - **Claim 1** (supported): The audited sources cover equal tau values, prescribed tau progressions, distinct omega values, and ratios of adjacent tau values; none supplies this bounded rainbow record. [4](#reference-4) [3](#reference-3) [1](#reference-1) [2](#reference-2) ### Runnable artifacts - **Artifact 1** (reproduced): A C++17 program reconstructs ten billion divisor counts, hashes their full stream, and retains sliding-window state across 9,537 segments. [4](#reference-4) ### Computational notes - A smallest-prime-factor sweep through 5000000 found maximum length 11, first attained at 590890. Earlier record lengths 8, 9, and 10 first began at 2204, 3718, and 95499. Every divisor count in the displayed interval was reconstructed from prime exponents. ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `rainbow-divisor-count-run-1e12`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1 entry has incomplete source metadata. Each affected row names the fields that still need editorial review. 1. Sean Eberhard, “Ratios of consecutive values of the divisor function,” Journal of Number Theory 281 (2026), 426-428. DOI 10.1016/j.jnt.2025.10.002; arXiv:2505.00727v2. theorems on ratios of consecutive values of the divisor function https://arxiv.org/abs/2505.00727 - Also cited at Sean Eberhard, Ratios of consecutive values of the divisor function, 2025 - preprint; reference source; arXiv:2505.00727, checked 2026-08-01; checked 2026-07-25 - Source use: citation_only - For Longest rainbow divisor-count interval below 10^12, this source supplies neighboring divisor-function literature; it does not determine the exact finite rainbow run. 2. Jon E. Schoenfield, “A363335: Irregular table read by rows: T(n,k) is the smallest m that has 2*n divisors and is at the beginning of a run of exactly k consecutive integers whose number of divisors increases by 2,” On-Line Encyclopedia of Integer Sequences, submitted May 29, 2023, checked 2026-08-01. entry definition, comments, and examples for prescribed consecutive tau values https://oeis.org/A363335 - Also cited at OEIS A363335, prescribed consecutive values tau(m+j)=2(n+j) - reference_database; reference source; checked 2026-07-25 - Source use: citation_only - Reused material: entry definition, comments, and examples for prescribed consecutive tau values - Reuse basis: fair_use_reviewed; rights holder: The OEIS Foundation Inc. and the credited contributors; checked 2026-08-01; by Philip Weiss, TheoremDB staff - Required attribution: Jon E. Schoenfield, “A363335: Irregular table read by rows: T(n,k) is the smallest m that has 2*n divisors and is at the beginning of a run of exactly k consecutive integers whose number of divisors increases by 2,” On-Line Encyclopedia of Integer Sequences, submitted May 29, 2023, checked 2026-08-01. - For Longest rainbow divisor-count interval below 10^12, this source records neighboring prescribed consecutive divisor-count patterns; it does not answer the packet's distinct-value optimization. 3. Jean-Marie De Koninck, John B. Friedlander, and Florian Luca, “On strings of consecutive integers with a distinct number of prime factors,” Proceedings of the American Mathematical Society 137(5) (2009), 1585-1592. DOI 10.1090/S0002-9939-08-09702-5. main theorems on runs with distinct omega and Omega values https://www.jeanmariedekoninck.mat.ulaval.ca/fileadmin/Documents/Publications/2009_on_string_of_consecutive_integers_with_a_distinct_number_of_prime_factors.pdf - Also cited at Jean-Marie De Koninck, John B. Friedlander, and Florian Luca, On strings of consecutive integers with a distinct number of prime factors, Proceedings of the AMS 137 (2009), 1585-1592 - website; reference source; checked 2026-07-25 - Source use: citation_only - For Longest rainbow divisor-count interval below 10^12, this source supplies an analogous result for distinct prime-factor counts, a different function from tau. 4. Vladimir A. Letsko, Some new results on consecutive equidivisible integers, 2015 https://arxiv.org/abs/1510.07081 - Also cited at Exact factorization table in rdcr-claim-factorized-run and exhaustive computation in rdcr-artifact-segmented-prefix-sweep - Also cited at Independent trial-division replay on 2026-07-25 and factor output from rdcr-artifact-segmented-prefix-sweep - Also cited at rdcr-artifact-segmented-prefix-sweep - Also cited at Vladimir A. Letsko, Some new results on consecutive equidivisible integers, 2015, especially the introduction and reported runs - Also cited at Inline C++17 and OpenSSL computation executed on 2026-07-25 - preprint; reference source; checked 2026-07-25 - Source metadata incomplete: publication-style citation, source version. - Source use: citation_only - Source named by the research packet.