At that point you’re really just exploring the integer multiples of 111, where it’s perhaps not surprising to find patterns in base 10, since 111 is 10^0+10^1+10^2, and two digit integers you multiply it by are going to be of the form A10^0+B10^1, so the resulting numbers are all going to be A10^0+(A+B)10^1+(A+B)10^2+B10^3. When A+B < 10, that’s going to form a pleasing pattern as you increment through As and Bs.

The pattern should actually be fine up to 37*54 = (37*3)*18 = 1998; it breaks at 37*57 because that’s (37*3)*19 and 1+9 isn’t less than 10.

Similarly interesting patterns will crop up when multiplying 1111 or 11111 by integers - and similarly in other bases as well (identically, in fact - multiples of 0x111 include 0x1221, 0x1332, etc)

So this sort of reduces to ‘I guess It’s funny that 111 is 37*3’.. but on some level, that’s just how numbers work. Every number has a prime factorization. 1111’s prime factors are 11 and 101. 0x111’s prime factors are 0x3, 0x7 and 0xD; 111’s just happen to be 3 and 37.

That said… it turns out 37 crops up as a prime factor of 111, 111111, 111111111, 111111111111… so there’s something a little more going on here…

Although even that is really just because each of those numbers is 111* a number of the form 1001…001001001 - 1000^0+1000^1+…+1000^n