Quadratic Friends (The Man Who Counted)

J1, J2, and I are currently reading The Man Who Counted. Here are some quick thoughts:

Quadratic Friends
The book is a great entry point for mathematical discussions. In fact, it makes it questionable as bedtime reading, since I have to be careful to find a more narrative section to close the evening. Otherwise, we would just continue talking and they'd never get to sleep.

Fortunately, the J's are willing to extend some of these conversations over to the next day, so we're not obligated to wrap up everything in one evening.

Here is an example discussion: in one of the early chapters, the protagonist Beremiz talks about the special relationship between 13 and 16. Namely:

13 * 13 = 169
1 + 6 + 9 = 16
16 * 16 = 256
2 + 5 + 6 = 13

Finding more
We wondered: what other pairs of numbers share this property?

Our first instinct was to gather data, so we started calculating some examples. We began with 0 and worked up, squaring, adding the digits, repeating. We found a couple of cases that flowed into the 13-16 relationship, for example 7. This gives a feeling that 7 is very fond of 13, but 13 only has eyes for 16.  Not the usual way people think about numbers, I guess.

Along the way, we made some interesting observations about this iterative process. I won't spoil the surprise, but would encourage you to explore yourself.

I'd note that J1 did the calculations up to 30 in his head, while I was a bit lazy and wrote a pencilcode program.

An extension
This conversation branched in an interesting way. Squaring is a natural thing to do with numbers, but summing the digits is a bit artificial. It depends on a choice of base. So, a natural follow-up question:
what quadratic friends exist in other bases?  This is an exploration for another day.