523776, 523777 is another example of a 2-run. Again, 523776 is a triperfect number.

An example of a 3-run is 5829840, 5829841, 5829842. No apparent pattern in these three numbers:

```
Factorization Sum of proper divisors
5829840 2^4 * 3^3 * 5 * 2699 14258160
5829841 29 * 53 * 3793 316439
5829842 2 * 2914921 2914924
```

Here are two more perfect 3-runs of consecutive numbers:

```
Factorization Sum of proper divisors
3414097920 2^16 * 3 * 5 * 23 * 151 8061430272
3414097921 13 * 31 * 41 * 206627 473814527
3414097922 2 * 1707048961 1707048964
39339578248 2^3 * 4917447281 34422130982
39339578249 29 * 4561 * 297421 1365596671
39339578250 2 * 3^2 * 5^3 * 7 * 2497751 82231007094
```

The first of these two, like the earlier example, were found by computer search with PARI/GP. I noticed that their middle odd terms turn up in A072188:

Numbers $n$ such that $n$ divides $\sigma(n-1)+\sigma(n)+\sigma(n+1)$, where $\sigma(n)$ is the sum of integer divisors of $n$.

The final of the above examples of perfect 3-runs comes from that sequence.