# E059: Abundancy index landscape

```{figure} ../_static/experiments/e059_hero.png
:width: 80%
:alt: Preview figure for E059
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `arithmetic-functions`, `sigma`, `divisor-function`, `perfect`
See: {doc}`../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)

- {doc}`../background/divisor-functions`
- {doc}`../background/perfect-numbers`

## 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).

```{include} ../reports/e059.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

::: {dropdown} params.json (snapshot)
:open:

```{literalinclude} ../params/e059.json
:language: json
```

:::

## References

See {cite:t}`apostol1976introanalyticnumbertheory`, {cite:t}`nivenzuckermanmontgomery1991introductiontheorynumbers`.

## Related experiments

- {doc}`e097` (E097: σ(n)/n landscape: deficient, perfect, abundant)
- {doc}`e003` (Abundancy Index Landscape)
- {doc}`e058` (Divisor-count record highs)
- {doc}`e096` (E096: Record-holders for τ(n))
- {doc}`e052` (Totient ratio landscape)