# E006: Near Misses to Perfection

```{figure} ../_static/experiments/e006_hero.png
:width: 80%
:alt: Preview figure for E006
```

```{figure} ../_static/experiments/e006_hero_2.png
:width: 80%
:alt: Preview figure for E006
```

**Tags:** `number-theory`, `conjecture-generation`, `visualization`, `numerics`
See: {doc}`../tags`.

## Highlights

- Search for $n$ with $\sigma(n)$ unusually close to $2n$.
- Build leaderboards for absolute and relative deviation.
- Visualize how “near perfection” clusters by structure.


## Goal

Find and visualize integers whose divisor sum is **unusually close** to the perfect condition $\sigma(n)=2n$,
without being perfect.

## Research question

For integers $n \le N$, which numbers minimize:

- absolute deviation:
  $$
  D_1(n) = |\sigma(n)-2n|
  $$
- relative deviation:
  $$
  D_2(n) = \left|\frac{\sigma(n)}{n}-2\right|?
  $$

Do “near misses” cluster in recognizable families (highly composite, abundant, etc.)?

## Why this qualifies as a mathematical experiment

The perfect condition is a sharp equality. Studying the closest failures often reveals structure and suggests
new questions (e.g., which multiplicative patterns drive $\sigma(n)$ toward $2n$).

## Experiment design

### Computation

- Compute $\sigma(1..N)$ via the divisor-sum sieve (as in E003).
- For each $n$, compute $D_1(n)$ and $D_2(n)$.
- Keep the top-$k$ smallest deviations (excluding actual perfect numbers).

### Outputs

- table: top-$k$ near misses (with $n$, $\sigma(n)$, $D_1$, $D_2$)
- plot: $n$ vs. $D_2(n)$ (log-scale on $D_2$ often helps)
- mark perfect numbers for reference

## How to run

```bash
make run EXP=e006
```

or:

```bash
uv run python -m mathxlab.experiments.e006
```

## Notes / pitfalls

- Use integer comparisons to identify perfect numbers (`sigma[n] == 2*n`).
- For $D_2$, floats are fine for plotting, but store exact rational values for ranking when possible
  (e.g., compare $|\sigma(n)-2n|$ first, then normalize for reporting).
- Choose $k$ small (e.g. 50 or 200) so the report stays readable.

## Extensions

- Repeat for different $N$ and compare stability of the “near miss” leaderboard.
- Add a second leaderboard restricted to odd $n$ only.
- Compare near misses to known abundant/deficient classifications and prime factorizations.

## Published run snapshot

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

```{include} ../reports/e006.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e006.json
:language: json
```

:::

## References

See {doc}`../references`.

{cite:p}`OEIS2025A000396PerfectNumbers,Voight1998PerfectNumbersElementaryIntroduction,Weisstein2003PerfectNumberMathWorld`

## Related experiments

- {doc}`e027` (Record prime gaps vs. log^2 heuristic)
- {doc}`e029` (Twin primes: observed vs. heuristic)
- {doc}`e086` (E086: Hardy Z(t) near zeros)
- {doc}`e002` (Even Perfect Numbers — Generator and Growth)
- {doc}`e003` (Abundancy Index Landscape)