TITLE: Perfect Runs of Consecutive Integers
QUESTION [11 upvotes]: A run of 2 or more consecutive integers is said to be perfect if the sum of its terms equals the sum of all the their proper divisors, adding common divisors as often as they occur. For example, 672, 673 is such a run (672 is a triperfect number whose proper divisors add up to 1344, while 673 is a prime, 1 being its only proper divisor).
Are there arbitrarily long perfect runs of consecutive numbers? Infinitely many of any length greater than 1?

REPLY [6 votes]: 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.