E098: Maximizers of σ(n)/n^α across α

E098 hero

E098: Maximizers of σ(n)/n^α across α

Tags: number-theory, quantitative-exploration, visualization, sigma, optimization

Highlights

  • Focused numeric experiment with a small set of figures.

  • Parameters saved to params.json for reproducibility.

  • Defaults are chosen, so the experiment remains feasible for the CI “slow” suite.

What is computed

  • For α on a grid, find n≤N maximizing σ(n)/n^α and show regime changes.

Notes

  • This page summarizes the intent; see the generated report.md in out/ for concrete outputs.

Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

Artifacts

  • figures/fig_01_argmax_vs_alpha.png

  • params.json

  • report.md

Notes

  • Unique maximizers encountered: 7

  • First transitions (alpha, argmax n): [(0.5, 55440), (1.0384615384615385, 5040), (1.076923076923077, 120), (1.1153846153846154, 12), (1.2307692307692308, 6), (1.2692307692307692, 2), (1.6153846153846154, 1)]

  • This is a small-N proxy for highly/colossally abundant phenomena.

params.json (snapshot)
{
  "alpha_max": 2.0,
  "alpha_min": 0.5,
  "alpha_steps": 40,
  "n_max": 60000
}

References

  • See the arithmetic-functions background pages in docs/background/.