Divisor Count
number-theory · 37 views
number-theoryarithmetic
What It Is
Divisor count d(n) --- erratic arithmetic function averaging ~ln(n), spikes at highly composite numbers, modulated by prime factorization structure
Interpretation
Standard analysis sees: heavy-tailed; right-skewed; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power). The atlas finds no named structure, but the source is distinctively extreme on Septagonal (Danzer):z_primary (-2.6z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.92
asymmetry0.94
occupancy0.20
short-range corr0.13
long-range memory0.09
spectral colour0.89
periodicity0.68
complexity0.74
time-irreversibility0.45
volatility clustering0.35
multifractality0.44
dimensionality0.53
nonstationarity0.21
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
Septagonal (Danzer):z_primary | -2.6z | bank-miss 1.0σ |
Septagonal (Danzer):cubic_coherence | -2.2z | bank-miss 1.3σ |
Composition
dtypefloat64
range[2, 180]
unique values51 / 16384
mean ± std14.1 ± 15.2
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > free_plate_nodal
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Divisor_Count.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Divisor_Count.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Divisor_Count.png)
Inflation (Substitution) > barcode
/d_curve/Divisor_Count.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/Divisor_Count.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/Divisor_Count.png)
Ulam Spiral (Sacks) > default
/default/Divisor_Count.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Prime Gaps | 3.76 | |
| Aliquot Orbit Lengths | 3.88 | |
| Collatz Flights | 4.04 |
Which Geometries Light Up
2-adic › 2-adic:valuation_transition_predictability | rank 303/306 | 0.3507 |
Zipf–Mandelbrot (16-bit) › Zipf–Mandelbrot (16-bit):zipf_r_squared | rank 4/306 | 0.9877 |
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