E059: Abundancy index landscape

Preview figure for E059

Tags: number-theory, quantitative-exploration, visualization, arithmetic-functions, sigma, divisor-function, perfect See: Valid Tags.

Highlights

  • Plot \(\sigma(n)/n\) and mark the threshold 2 (perfect numbers).

  • Highlight spikes caused by many small prime factors.

Goal

Connect a divisor-sum function to the perfect/abundant classification visually.

Background (quick refresher)

Research question

What shapes and spikes appear in \(\sigma(n)/n\) over moderate ranges, and where do perfect numbers sit?

Method

  • Compute \(\sigma(n)\) up to \(N\) using factorization via SPF.

  • Plot \(\sigma(n)/n\) and annotate known perfect numbers in range.

How to run

  • make run EXP=e059

  • uv run python -m mathxlab.experiments.e059

Outputs

This experiment follows the standard output contract:

  • out/e059/figures/ — generated figures (PNG)

  • out/e059/report.md — short narrative report

  • out/e059/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: 200000

Top 12 values of σ(n)/n (value, n):

  • 4.29437 at n=166320

  • 4.25455 at n=110880

  • 4.23932 at n=196560

  • 4.20000 at n=131040

  • 4.18701 at n=138600

  • 4.18701 at n=55440

  • 4.15584 at n=83160

  • 4.13333 at n=163800

  • 4.13333 at n=151200

  • 4.13333 at n=65520

  • 4.12941 at n=171360

  • 4.12430 at n=194040

Figure:

  • fig_01_sigma_over_n.png

params.json (snapshot)
{
  "n_max": 200000,
  "top_k": 12
}

References

See Apostol [1976], Niven et al. [1991].