[#P2430] Merging of orbits under adding the product of nonzero digits

Problem. Define \(T(n) = n + p(n)\), where \(p(n)\) is the product of the nonzero decimal digits of \(n\). Do all orbits of \(T\) eventually merge into finitely many trajectories, in the sense that there is a finite set \(S\) of orbits such that every starting value eventually enters one of them?

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1Context

Multiples of 10 with a single nonzero digit behave differently from generic values, since p(n) is then small relative to n, which is the obvious obstruction to a clean merging argument.

2Remarks

Remark 1. The product of nonzero digits ignores any digit equal to 0, so p(105) = 1 times 5 = 5.

Remark 2. Two orbits merge when they share a common value from some point on.

3What counts as a solution

Prove that only finitely many distinct eventual trajectories occur, or show that infinitely many pairwise disjoint orbits exist.

1Status

What counts as a solution

Current status (Loomis conjectured that every orbit joins the orbit of 1). The published conjecture gives a single eventual trajectory, which is stronger than the candidate's finite-trajectory question.^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
OEIS A063114 records the map, while A063108 records the orbit of 1.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 OEIS A063114 records the map, while A063108 records the orbit of 1.^[2] Relevance to this problem For Merging of orbits under adding the product of nonzero digits, record dpi-claim-oeis-identification (“The map and principal orbit have OEIS records”) records a bound, answer, status fact, or structural consequence. The record states: OEIS A063114 records the map, while A063108 records the orbit of 1. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope the map and the orbit beginning at 1 Sources Reference 2 Argument OEIS A063114 is the transformation \(n\mapsto n+p(n)\). OEIS A063108 is its orbit beginning at 1, and A096287 records joining times for individual starts. These records cite Loomis and preserve the conjectural status of universal joining. Related: Conjecture 1 Open full record →
The published conjecture gives a single eventual trajectory, which is stronger than the candidate's finite-trajectory question.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Conjecture 1 The published conjecture gives a single eventual trajectory, which is stronger than the candidate's finite-trajectory question.^[1] Relevance to this problem For Merging of orbits under adding the product of nonzero digits, record dpi-claim-loomis-conjecture (“Loomis conjectured that every orbit joins the orbit of 1”) records a bound, answer, status fact, or structural consequence. The record states: The published conjecture gives a single eventual trajectory, which is stronger than the candidate's finite-trajectory question. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state conjectured Scope every positive starting value under the digit-product map Sources Reference 1 Details For \(T(n)=n+p(n)\), with zero digits omitted from the product, Paul Loomis conjectured that every positive starting value eventually enters the orbit beginning at 1. This exact map and conjecture appeared in his article on digit product sequences and on the companion page's section "Sequences That Join." The candidate independently rediscovered a weaker consequence. The reviewed sources still describe universal joining as a conjecture. Open full record →
Universal joining to the orbit of 1 implies the candidate, while the weaker finite-trajectory statement may admit a separate proof.	Trace	Evidence: Conjectured Proposed as likely true. No argument is recorded.
attempt · Route 1 Universal joining to the orbit of 1 implies the candidate, while the weaker finite-trajectory statement may admit a separate proof.^[2] Relevance to this problem For Merging of orbits under adding the product of nonzero digits, record dpi-attempt-finite-versus-single-orbit (“Separate the finite-orbit question from Loomis's conjecture”) documents a concrete method, search boundary, or failed route. The record states: Universal joining to the orbit of 1 implies the candidate, while the weaker finite-trajectory statement may admit a separate proof. Evidence ConjecturedProposed as likely true. No argument is recorded. Record state open strategy Scope No narrower scope recorded Sources Reference 2 What happened The literature audit found no independent argument that only finitely many disjoint eventual trajectories exist. One route is to bound the number of disjoint increasing orbits using decimal intervals and the small increments near powers of ten. The opposite route would construct infinitely many starts whose forward sets stay disjoint. A proof of Loomis's stronger conjecture would settle both questions at once. Related: Conjecture 1 Open full record →
Every start through one million was reported to join the orbit of 1.	Artifact	Evidence: Supported Backed by a cited source or by evidence short of a proof.
artifact · Artifact 1 Every start through one million was reported to join the orbit of 1.^[3] Relevance to this problem For Merging of orbits under adding the product of nonzero digits, record dpi-artifact-million-starts (“Loomis reports a million-start joining check”) supplies evidence or a replay used to check the packet. The record states: Every start through one million was reported to join the orbit of 1. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state sourced Scope every starting value from 1 through 1000000 Sources Reference 3 Details Loomis reports testing every positive start at most one million. Each orbit joined the orbit beginning at 1. The first difficult start was 63, with the recorded identity \[ T^{322}(63)=150056=T^{261}(1). \] This is author-reported computation rather than a retained TheoremDB program. Related: Conjecture 1 Open full record →
The 199 starts produce 31 distinct values after exactly 60 applications of the map.	Artifact	Evidence: Computational evidence The record reports a computation within its stated scope. Reproduction requires a separate replay.
artifact · Artifact 2 The 199 starts produce 31 distinct values after exactly 60 applications of the map. Relevance to this problem For Merging of orbits under adding the product of nonzero digits, record dpi-artifact-60-step-endpoints (“Sixty iterations leave 31 distinct endpoints”) supplies evidence or a replay used to check the packet. The record states: The 199 starts produce 31 distinct values after exactly 60 applications of the map. Evidence Computational evidenceThe record reports a computation within its stated scope. Reproduction requires a separate replay. Record state available Scope starts 1 through 199, each iterated exactly 60 times File research/candidate-screening/multi.py Details The original candidate said the runs "produced only 31 distinct values," though the 199 starting values are already distinct. The code measures the endpoints after exactly 60 steps. Its correct result is that \(\{T^{60}(s):1\leq s\leq199\}\) has size 31. Equal endpoints certify that the corresponding finite orbits have merged by that time. Related: Conjecture 1 Open full record →
No records match these filters.

Notes and companion material

Original intake status. UNKNOWN as of 2026-07-28. Loomis conjectures the stronger statement that every starting value eventually joins one canonical trajectory. The checked computation verifies this through one million, without a proof for all integers.

The 2026-07-28 audit found a stronger published conjecture: every orbit joins the orbit beginning at 1.
The packet and source record retain verification through one million as bounded evidence only.
No duplicate digit-product coalescence target was found in the controlled corpus.

Recorded example 1. The orbit of 1 begins 1, 2, 4, 8, 16, 22, 26, 38, 62, 74, 102, 104, 108, 116, 122.

Computational notes

Iterating 60 steps from each of the 199 starting values 1 through 199 produced only 31 distinct values, so the great majority of these orbits had already coalesced.

How the 5 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemMerging of orbits under adding the product of nonzero digits

All 4 recorded relations between these records and the problem

2See also

Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory
Square-class collisions in the Pell-Lucas sequencenumber theory

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Cite the original sources separately.

Plain text

“Merging of orbits under adding the product of nonzero digits.” TheoremDB. P2430. Problem statement; statement text SHA-256 2f4341e41a0cb1909c6f7781cd7280e4ea46bebc43e5be48f88879f09ec3c8dd. https://theoremdb.org/statement/?ref=P2430

BibTeX

@misc{theoremdb-problem-2f4341e41a0cb1909c6f7781cd7280e4ea46bebc43e5be48f88879f09ec3c8dd,
  title = {{Merging of orbits under adding the product of nonzero digits}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 2f4341e41a0cb1909c6f7781cd7280e4ea46bebc43e5be48f88879f09ec3c8dd},
  url = {https://theoremdb.org/statement/?ref=P2430}
}

Plain text: Built Markdown snapshot

This problem includes 5 records joined by 4 typed links, sourced from plouffe.fr^[1], current as of July 24, 2026.

1Lean verification

1References

Packet source. Paul A. Loomis, An Introduction to Digit Product Sequences, Journal of Recreational Mathematics 32 (2003-2004), pages 147-151; companion page section III. Digit-product iteration and the joining conjecture. ↗scholarly publication · reference source · PDF checked 2026-08-01 · checked 2026-08-01Source use: citation only.Introduces the digit-product iteration and conjectures that every start joins the orbit of 1.Also cited at Paul A. Loomis, An Introduction to Digit Product Sequences, Journal of Recreational Mathematics 32 (2003-2004), pages 147-151; companion page section III.For Merging of orbits under adding the product of nonzero digits: The published conjecture gives a single eventual trajectory, which is stronger than the candidate's finite-trajectory question.Source named by the research packet.Introduces the exact map and proposes a stronger one-trajectory version of this problem.
Open direction relative to the conjecture recorded by Loomis and OEIS A063108. OEIS entry A063108, checked 2026-08-01. Main entry and cross-references A063114 and A096287. ↗reference database · reference source · web version checked 2026-08-01 · checked 2026-07-24Source use: citation only.Records the canonical digit-product trajectory and its linked map and joining data.Also cited at Open direction relative to the conjecture recorded by Loomis and OEIS A063108.Also cited at OEIS A063108, A063114, and A096287.For Merging of orbits under adding the product of nonzero digits: Universal joining to the orbit of 1 implies the candidate, while the weaker finite-trajectory statement may admit a separate proof.Maintains the canonical trajectory, terminology, and current computational references.
Paul Loomis, Integer Sequences, section III, Sequences That Join; cross-referenced by OEIS A096287. facstaff.bloomu.edu checked 2026-08-01. Section III and the verification report. ↗website · reference source · web version checked 2026-08-01 · checked 2026-07-24Source use: citation only.Reports that every starting value through one million joins the canonical digit-product trajectory.Also cited at Paul Loomis, Integer Sequences, section III, Sequences That Join; cross-referenced by OEIS A096287.For Merging of orbits under adding the product of nonzero digits: Reports the checked finite verification that starting values through one million join the canonical trajectory.Reports the checked finite verification that starting values through one million join the canonical trajectory.

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[#P2430] Merging of orbits under adding the product of nonzero digits

1Context

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3What counts as a solution

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Computational notes

2See also

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