# E058: Divisor-count record highs

```{figure} ../_static/experiments/e058_hero.png
:width: 80%
:alt: Preview figure for E058
```

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

- {doc}`../background/divisor-functions`
- {doc}`../background/arithmetic-functions`

## 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=e058`
- `uv 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 report
- `out/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).

```{include} ../reports/e058.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e058.json
:language: json
```

:::

## References

See {cite:t}`ramanujan1915highlycompositenumbers`, {cite:t}`tenenbaum2015introanalyticprobabilisticnumbertheory`.

## Related experiments

- {doc}`e096` (E096: Record-holders for τ(n))
- {doc}`e059` (Abundancy index landscape)
- {doc}`e119` (E119: Summatory totient Φ(x) scaling check)
- {doc}`e117` (E117: Prime-counting approximations: li(x) and friends)
- {doc}`e123` (E123: π(x;q,a) vs. a simple baseline)