[#P2658] Longest rainbow divisor-count interval below 10^12

A mathematical schematic of Longest rainbow divisor-count interval below 10^12. — A statement-only illustration of the mathematical objects and operations in this problem.

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.

1Context

This finite search asks for the longest consecutive interval below the cutoff whose integers all have different divisor counts.

2Definitions

Definition 1 (divisor function). The divisor function \(\tau(n)\) is the number of positive divisors of \(n\).

Definition 2 (rainbow interval). A rainbow interval is a consecutive interval on which the divisor-count values are pairwise distinct.

3What 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.

1Status

Current status (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.^[1]

1Records

Record	Kind	Assessment
	Result	Reproduced
claim · Computation 1 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.^[1] Relevance to this problem For Longest rainbow divisor-count interval below 10^12, record rdcr-claim-global-lower-bound-fourteen (“The requested maximum is at least fourteen”) records a bound, answer, status fact, or structural consequence. The record states: 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. Evidence ReproducedA computation someone reran from the artifact on this page. Record state supported Scope the optimization over positive m and L satisfying m+L-1 <= 1000000000000 Sources Reference 1 Details 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 \(10^{10}\). Open full record →
	Result	Established
claim · Theorem 1 Trial division through the square root gives complete prime factorizations and reproduces all fourteen divisor counts.^[1] Relevance to this problem For Longest rainbow divisor-count interval below 10^12, record rdcr-claim-factorized-run (“Exact factorizations certify the length-fourteen witness”) records a bound, answer, status fact, or structural consequence. The record states: Trial division through the square root gives complete prime factorizations and reproduces all fourteen divisor counts. Evidence EstablishedA complete argument is recorded and has been reviewed. Scope the fourteen integers from 1745175039 through 1745175052 Sources Reference 1 Argument The complete factorizations are \[ \begin{array}{rcl} 1745175039&=&3\cdot31^2\cdot605333,\\ 1745175040&=&2^9\cdot5\cdot7\cdot97387,\\ 1745175041&=&37\cdot103\cdot173\cdot2647,\\ 1745175042&=&2\cdot3^2\cdot13\cdot19\cdot59\cdot6653,\\ 1745175043&=&53\cdot977\cdot33703,\\ 1745175044&=&2^2\cdot436293761,\\ 1745175045&=&3\cdot5\cdot41\cdot773\cdot3671,\\ 1745175046&=&2\cdot872587523,\\ 1745175047&=&7\cdot11\cdot61^2\cdot6091,\\ 1745175048&=&2^3\cdot3\cdot23\cdot47\cdot137\cdot491,\\ 1745175049&=&1745175049,\\ 1745175050&=&2\cdot5^2\cdot29\cdot167\cdot7207,\\ 1745175051&=&3^4\cdot113\cdot190667,\\ 1745175052&=&2^2\cdot17^2\cdot1039\cdot1453. \end{array} \] Trial division by every prime through the square root leaves either 1 or the displayed final prime, which proves each factorization complete. Applying \(\prod(e+1)\) gives, in the same order, \[ 12,80,16,96,8,6,32,4,24,128,2,48,20,36. \] Related: Computation 1 Open full record →
	Result	Reproduced
claim · Computation 2 A complete sweep of all ten billion divisor counts proves that no length-15 rainbow interval ends by 10,000,000,000.^[1] Relevance to this problem For Longest rainbow divisor-count interval below 10^12, record rdcr-claim-exact-prefix-maximum-fourteen (“The exact maximum through ten billion is fourteen”) records a bound, answer, status fact, or structural consequence. The record states: A complete sweep of all ten billion divisor counts proves that no length-15 rainbow interval ends by 10,000,000,000. Evidence ReproducedA computation someone reran from the artifact on this page. Record state established Scope every interval of consecutive positive integers whose final term is at most 10000000000 Sources Reference 1 Details For a segment \([a,b]\), the program initializes a residual copy of every integer and a divisor count equal to 1. For each prime \(p\leq\sqrt b\), it removes the full power \(p^e\) from every multiple of \(p\) in the segment and multiplies the divisor count by \(e+1\). A residual factor greater than 1 is prime and contributes a final factor 2. This reconstructs \[ \tau(n)=\prod_{p^e\parallel n}(e+1) \] exactly for every integer in the segment. The sliding search stores the most recent position of each divisor count. Immediately before processing \(n\), let \(s\) begin the current rainbow suffix. After reading \(d=\tau(n)\), replace \[ s\leftarrow\max\{s,\operatorname{last}(d)+1\}. \] Then \(n-s+1\) is the longest rainbow interval ending at \(n\). Both \(s\) and the last-position table persist across segment boundaries. Maximizing this value over all \(1\leq n\leq10^{10}\) gives 14, first at the displayed witness. This proves the prefix maximum and rules out every length-15 interval ending in that checked range. Open full record →
	Result	Supported
claim · Claim 1 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.^[1][4][2][3] Relevance to this problem For Longest rainbow divisor-count interval below 10^12, record rdcr-claim-literature-audit (“Nearby literature treats different consecutive-divisor questions”) records a bound, answer, status fact, or structural consequence. The record states: 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. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope a targeted literature and sequence-database search for consecutive integers with pairwise distinct values of the ordinary divisor function, checked through 2026-07-25 Sources Reference 1, Reference 4, Reference 2, Reference 3 Details Letsko studies consecutive integers having one fixed value of \(\tau\), including long runs with 12 or 24 divisors. De Koninck, Friedlander, and Luca prove existence bounds for consecutive integers whose \(\omega\) or \(\Omega\) values are pairwise distinct. Those functions count prime factors rather than divisors. Eberhard proves that every positive rational occurs infinitely often as \(\tau(n+1)/\tau(n)\), a result about adjacent pairs. OEIS A363335 catalogs runs satisfying the prescribed pattern \(\tau(m+j)=2(n+j)\). Its rows give examples with distinct divisor counts under an extra linear constraint. A targeted search of these papers, their references, arXiv, and OEIS did not locate a published maximum for arbitrary pairwise-distinct \(\tau\)-values below \(10^{12}\). This is a scoped status report rather than a proof of novelty. Related: Computation 1 Open full record →
	Artifact	Reproduced
artifact · Artifact 1 A C++17 program reconstructs ten billion divisor counts, hashes their full stream, and retains sliding-window state across 9,537 segments.^[1] Relevance to this problem For Longest rainbow divisor-count interval below 10^12, record rdcr-artifact-segmented-prefix-sweep (“Segmented factor sieve and cross-segment rainbow search”) supplies evidence or a replay used to check the packet. The record states: A C++17 program reconstructs ten billion divisor counts, hashes their full stream, and retains sliding-window state across 9,537 segments. Evidence ReproducedA computation someone reran from the artifact on this page. Record state available Scope all divisor counts tau(n) for 1 <= n <= 10000000000 and every rainbow interval ending in that range Sources Reference 1 Run `c++ -O3 -std=c++17 -march=native -I/opt/homebrew/include -L/opt/homebrew/lib sweep.cpp -lcrypto -o sweep && ./sweep 10000000000` Runtime C++17 with unsigned 64-bit integers and OpenSSL libcrypto on a little-endian host Details Compile the source with the command shown and run it with limit `10000000000`. The program uses 64-bit residuals and 16-bit divisor counts in blocks of \(2^{20}\) integers. The prime table contains all 9,592 primes through 100,000. The final deterministic fields are ``` BEST limit=10000000000 length=14 start=1745175039 end=1745175052 CERT tau_u16le_sha256=2cc2b23f733074b8177d70dbb84fc4c14e2d57702748fd32b31c33e66fcecafb tau_sum=231802823220 tau_square_sum_mod_2^64=16509952757456 primes=9592 block=1048576 ``` The run also prints each new record and the complete factorization of every term in the final record. Its measured wall time on the research host was 107.508 seconds. The SHA-256 digest of the newline-terminated source snapshot is `02f5ddc8ca3d5d8792af25aec5b40cabb2d0f65c5040c54166781e2f8728075d`. Related: Computation 2 · Theorem 1 Open full record →
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Notes and companion materialContext, examples, and computations

Original intake status. OPEN in the reviewed TheoremDB packet as of 2026-08-01. 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.

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.
Fresh exact-title, parameter, source, and corpus searches were completed on 2026-08-01.

Recorded example 1. The interval 590890 through 590900 has divisor counts 16,12,24,6,8,20,10,4,32,2,36, all distinct.

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.

How the 5 records connectTyped relations and evidence flow

ProblemLongest rainbow divisor-count interval below 10^12

Computation 1The requested maximum is at least fourteenin this packetReproduced
Theorem 1Exact factorizations certify the length-fourteen witnesssupportsEstablished
Artifact 1Segmented factor sieve and cross-segment rainbow searchsupportsReproduced
Claim 1Nearby literature treats different consecutive-divisor questionsinformsSupported

1 record with no typed link to the problem

Computation 2The exact maximum through ten billion is fourteen

2See also

Largest rainbow squarefree gap below 10^12multiplicative number theory
Most divisors of a binomial coefficient with top at most 10^6multiplicative number theory

How to cite

TheoremDB contributors, “Longest rainbow divisor-count interval below 10^12,” TheoremDB research memory, snapshot of July 25, 2026. https://theoremdb.org/statements/rainbow-divisor-count-run-1e12

This page as plain text: rainbow-divisor-count-run-1e12.md

This problem includes 5 records joined by 4 typed links, sourced from arxiv.org^[1], current as of July 25, 2026.

1References

Packet source. arXiv preprint 1510.07081, linked source for Longest rainbow divisor-count interval below 10^12. Source location cited by the reviewed packet record. ↗preprint · primary source · arXiv:1510.07081, checked 2026-08-01 · checked 2026-07-25Source use: original summary.Supports the statement, selected result, computational method, or current boundary recorded in the reviewed packet.Also cited at Vladimir A. Letsko, Some new results on consecutive equidivisible integers, 2015.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.Source named by the research packet.
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. ↗preprint · reference source · arXiv:2505.00727, checked 2026-08-01 · checked 2026-07-25Source 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.Also cited at Sean Eberhard, Ratios of consecutive values of the divisor function, 2025.
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. ↗reference database · reference source · checked 2026-07-25Source 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.Also cited at OEIS A363335, prescribed consecutive values tau(m+j)=2(n+j).
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. ↗website · reference source · checked 2026-07-25Source 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.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.

CC0 bounded search for diverse divisor-count patterns.

[#P2658] Longest rainbow divisor-count interval below 10^12

1Context

2Definitions

3What counts as a solution

1Status

1Records

Computational notes

2See also

1Lean verification

1References

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Definitions

What counts as a solution

What's known

Formal statement

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1Context

2Definitions

3What counts as a solution

1Status

1Records

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance