E058: Divisor-count record highs¶
Tags: number-theory, quantitative-exploration, visualization, arithmetic-functions, divisor-function, bounds
See: Valid Tags.
Highlights¶
Track record highs of \(\tau(n)\) as \(n\) grows.
Inspect factorizations of record-holders (many small primes with decreasing exponents).
Goal¶
Make the record phenomenon behind highly composite numbers tangible.
Background (quick refresher)¶
Research question¶
How do record values of \(\tau(n)\) grow, and what structure do record-holders share?
Method¶
Compute \(\tau(n)\) up to \(N\) and track record positions.
Plot record curve; list factorizations of top records.
How to run¶
make run EXP=e058uv run python -m mathxlab.experiments.e058
Outputs¶
This experiment follows the standard output contract:
out/e058/figures/— generated figures (PNG)out/e058/report.md— short narrative reportout/e058/manifest.json— snapshot metadata for the gallery
Published run snapshot¶
If this experiment is included in the docs gallery, include the published snapshot (report + params).
n_max: 300000
max τ(n) in range: 180 at n=277200
Figure:
fig_01_tau_records.png
params.json (snapshot)
{
"n_max": 300000
}