Aliquot Orbit Lengths
number_theory · 36 views
number_theory
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 entries cap at the divergence ceiling --- treated as MAX_STEPS, not specially flagged
Interpretation
Standard analysis sees: right-skewed; anti-correlated (alternating); homoskedastic. The atlas additionally detects critical slowing down.
What standard analysis sees
tail heaviness0.79
asymmetry0.87
occupancy0.24
short-range corr0.07
long-range memory0.20
spectral colour0.71
periodicity0.81
complexity0.51
time-irreversibility0.36
volatility clustering0.10
multifractality0.33
dimensionality0.55
nonstationarity0.49
What the atlas adds
critical slowing down+5.9z
lag-1 autocorrelation rises across the series — the early-warning signature of an approaching tipping point / regime shift (also fired by frequency chirps)
also fires on nonstationary arithmetic sequences whose autocorrelation drifts for non-dynamical reasonsAtlas-extreme metrics the standard bank can’t predict for this source
Heisenberg (Nil) (centered):xy_spread | +3.8z | bank-miss 1.8σ |
Moiré:moire_peak_alpha | +2.8z | bank-miss 1.5σ |
Composition
dtypefloat64
range[1, 69]
unique values63 / 16384
mean ± std10.1 ± 8.83
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Aliquot_Orbit_Lengths.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Aliquot_Orbit_Lengths.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Aliquot_Orbit_Lengths.png)
Inflation (Substitution) > barcode
/d_curve/Aliquot_Orbit_Lengths.png)
Inflation (Substitution) > d_curve
Information Theory > default
Julia Set > escape_histogram
Klein Bottle > default
Laplacian > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Ordinal Partition > default
/phi_spectrum/Aliquot_Orbit_Lengths.png)
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > h1_diagram
Predictability > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > phase_space
/default/Aliquot_Orbit_Lengths.png)
Ulam Spiral (Sacks) > default
/default/Aliquot_Orbit_Lengths.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Divisor Count | 3.94 | |
| Prime Gaps | 4.13 | |
| Collatz Flights | 4.20 |
Which Geometries Light Up
H4 600-Cell › H4 600-Cell:lattice_closure | rank 3/298 | 0.9995 |
Moiré › Moiré:moire_peak_alpha | rank 3/298 | 3.4000 |
Nonstationarity › Nonstationarity:ac1_trend | rank 2/298 | 0.4348 |
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