Aliquot Orbit Lengths
What It Is
Aliquot sequence orbit lengths --- number of steps before s(n)=sigma(n)-n iteration terminates (reaches 1, enters a cycle, or escapes the sieve). Distinct erratic arithmetic from Collatz Stopping Times: divisor-sum dynamics instead of 3n+1 parity. Catalan-Dickson conjecture (some orbits unbounded, e.g. n=276) means a few orbits don't terminate; in practice they exit at ~20-25 steps when σ(n) escapes the SIGMA_MAX sieve (well short of the MAX_STEPS cap), so divergent orbits appear as ordinary small counts, not a MAX_STEPS ceiling
Interpretation
Standard analysis sees: right-skewed; anti-correlated (alternating); homoskedastic. The atlas finds no named structure, but the source is distinctively extreme on Nonstationarity:variance_trend (+2.7z) — beyond what the standard bank predicts for it.
Nonstationarity:variance_trend | +2.7z | bank-miss 1.4σ |
Moiré:moire_peak_alpha | +2.7z | bank-miss 1.3σ |
Composition
Render Gallery
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/phi_spectrum/Aliquot_Orbit_Lengths.png)
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Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Divisor Count | 3.87 | |
| Prime Gaps | 3.97 | |
| Collatz Flights | 4.01 |
Which Geometries Light Up
H4 600-Cell › H4 600-Cell:lattice_closure | rank 3/307 | 0.9995 |
Moiré › Moiré:moire_peak_alpha | rank 4/307 | 3.4000 |
Nonstationarity › Nonstationarity:ac1_trend | rank 2/307 | 0.4348 |
Nonstationarity › Nonstationarity:variance_trend | rank 5/307 | 0.2995 |