# E003: Abundancy Index Landscape

```{figure} ../_static/experiments/e003_hero.png
:width: 80%
:alt: Preview figure for E003
```

```{figure} ../_static/experiments/e003_hero_2.png
:width: 80%
:alt: Preview figure for E003
```

```{figure} ../_static/experiments/e003_hero_3.png
:width: 80%
:alt: Preview figure for E003
```

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

## Highlights

- Compute $\sigma(1..N)$ via a divisor-sum sieve.
- Visualize the distribution of $I(n)=\sigma(n)/n$.
- Highlight perfect numbers as the razor-thin level set $I(n)=2$.


## Goal

Visualize how **rare** perfect numbers are by plotting the distribution of the **abundancy index**
$$
I(n) = \frac{\sigma(n)}{n}
$$
for integers $n \le N$.
Perfect numbers satisfy $I(n)=2$.

## Research question

For a given bound $N$:

- What does the empirical distribution of $I(n)$ look like?
- How frequently do we see values near $2$?
- Where do the perfect numbers appear in the landscape?

## Why this qualifies as a mathematical experiment

The divisor-sum function $\sigma(n)$ is highly structured but “spiky” and hard to intuit symbolically.
Computational sweeps reveal qualitative structure (clusters, gaps, tails) and put perfect numbers into context.

## Experiment design

### Method: compute $\sigma(1)$, …, $\sigma(N)$ via a divisor-sum sieve

Use the identity “each divisor contributes to its multiples”:

- initialize an array `sigma[0..N]` with zeros
- for each `d = 1..N`:
  - add `d` to `sigma[k]` for all multiples `k = d, 2d, 3d, ...`

This runs in about $O(N\log N)$ time and avoids per-number factorization.

### Observables

- $I(n) = \sigma(n)/n$
- distance to perfection: $|I(n)-2|$
- classification:
  - deficient: $I(n)<2$
  - perfect: $I(n)=2$
  - abundant: $I(n)>2$

### Plots

- histogram of $I(n)$ (or of $I(n)-1$)
- scatter plot of $n$ vs. $I(n)$ (optionally with log-scale on $n$)
- highlight perfect numbers

## How to run

```bash
make run EXP=e003
```

or:

```bash
uv run python -m mathxlab.experiments.e003
```

## Notes / pitfalls

- $I(n)$ is rational. For classification, compare integers using $\sigma(n)$ and $2n$ rather than floats.
- For plots, floats are fine, but compute the “perfect” condition as `sigma[n] == 2*n`.
- Start with $N \le 1\,000\,000$ to keep runtime and memory reasonable.

## Extensions

- Repeat for different ranges and overlay histograms.
- Plot the top-$k$ values of $I(n)$ and compare to known extremal families.
- Explore the “near misses” set (feeds directly into E006).

## Published run snapshot

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

```{include} ../reports/e003.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e003.json
:language: json
```

:::

## References

See {doc}`../references`.

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

## Related experiments

- {doc}`e002` (Even Perfect Numbers — Generator and Growth)
- {doc}`e052` (Totient ratio landscape)
- {doc}`e059` (Abundancy index landscape)
- {doc}`e094` (E094: ω(n) vs. Ω(n): Erdős–Kac normalization)
- {doc}`e097` (E097: σ(n)/n landscape: deficient, perfect, abundant)