What this research found
Which starting values push the Collatz iteration to new extremes? An exhaustive scan of every integer up to 100 million tabulated both families of record-setter — the longest total stopping time, meaning the most steps needed to reach 1, and the highest peak visited along the way — finding 60 of the former and 41 of the latter. The full sweep ran in 3.87 seconds of single-core compute, and every record inside the published reference range matched the corresponding OEIS integer sequences exactly.
- The slowest trajectory below 100 million starts at 63,728,127 and needs 949 steps to reach 1. It closes a list of 60 total-stopping-time record-setters over that range.
- The highest climb starts at 80,049,391, whose trajectory peaks at 2,185,143,829,170,100 before descending to 1. It closes a list of 41 maximum-height record-setters.
- All 44 stopping-time and 25 maximum-height reference terms below 1 million reproduced the published OEIS values exactly, with zero discrepancies. The 32 newly tabulated records above 1 million were re-derived by a separate cache-free kernel and agreed in every case.
- Record stopping times grow logarithmically in the starting value, fitting S ≈ 46.822 ln n − 122.850 with an R² of 0.952 — roughly 108 extra steps per decade.
- Record peak heights follow a power law with an exponent of 1.858 (R² = 0.987), staying strictly below the n² envelope across the whole range. That sub-quadratic growth leaves the largest observed peak about 4200 times below the signed 64-bit integer ceiling.
- Skipping even starting values speeds up the height search legitimately but silently corrupts the stopping-time list, because S(2k) = S(k) + 1 always beats S(k). The even values 6, 18, and 54 are genuine stopping-time records that such a shortcut would discard.
How it was done
Rather than walking all 100 million trajectories down to 1, the engine scanned starting values in ascending order and cut each walk short the moment the value dropped below its own starting point, at which stage the already-cached result for that smaller value completed the answer. Two flat NumPy arrays held the memoized results — a 16-bit stopping-time cache and a 64-bit height cache, the latter needed because a peak is a running maximum rather than an additive count — and the inner loop was compiled to machine code with Numba. That combination sustained about 26 million starting values per second on one core of an Intel Xeon at 2.30 GHz, with peak memory of 1129.78 MB. Correctness was established three ways: term-by-term comparison against bundled OEIS reference tables, independent recomputation of every new record by a cache-free direct-walk kernel, and spot checks of canonical values such as S(27) = 111 and H(703) = 250504.
Data sources
- OEIS A006877 / A006878 — total-stopping-time record-setters and their step counts, 44 reference terms below 1 million
- OEIS A084605 / A084606 — maximum-trajectory-height record-setters and their peaks, 25 reference terms below 1 million
Limitations
A finite scan characterizes behaviour only up to its bound and cannot prove the Collatz conjecture, and the sub-quadratic growth of peak heights is an empirical regularity over this range rather than a proven asymptotic bound. Memory is what limits going further: the caches need about 10 bytes per starting value, so a bound of 10^9 would require roughly 10 GB.
How this research was produced
K-Dense Web planned and ran this mathematics investigation end to end — gathering the sources, carrying out the analysis, producing the figures, and drafting the report. The full session transcript, including every intermediate step, is available to view.
