Romantic and powerful pair numbers

Amicable numbers are two numbers where the sum of each numbers proper divisors equals the other.

The smallest pair of such numbers is 220 and 284. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. The one can produce the other, both ways.

A friend is “one who is the other I, such as are 220 and 284”.

Pythagoras

This connection between the pair numbers tie them together in an amicable (friendly) way, perhaps closer, in a romantic way. Also, as they can produce each other, I think they are powerful too.

To find more such pairs, I wrote a small program that look for them all up to 100 000. I’m sure the program can be more efficient:

The list of pairs are then as follows, starting with 220 and 284:

220 and 284
1184 and 1210
2620 and 2924
5020 and 5564
6232 and 6368
10744 and 10856
12285 and 14595
17296 and 18416
63020 and 76084
66928 and 66992
67095 and 71145
69615 and 87633
79750 and 88730

Are there more?

Yes, there are! But how many we don’t really know, which is the reason amicable numbers is classified as an unsolved problem in mathematics. Are there infinitely many such numbers? What do you think?