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
The fact you have to lay it out like you did kind of points towards the pattern being the result of prime factorization not being as evident as you seem to think it is lol
Apologies if I came off as dismissive here - was not my intent at all - there’s obviously tons of fascinating little patterns to dig into here.
Like, factorizing numbers of the form 111…111 is a whole thing…
If there’s a composite number of digits in the number, then you can always factor it into a smaller 11…11 number, and a number consisting of 1s separated by repeating 0s - a 100…00100…001 type number.
And you can do the same to that number if it has a composite number of 1s in it (turns out the 1111 case is just a ‘number with a bunch of ones separated by sequences of no zeroes’).
And it’s not a coincidence that the numbers of ones in those factors match the prime factors of 12.
So numbers in this form tend to have factors like 11, 101, and 111… and of course that means they tend to have 37 as a factor as a result. Like, in the above example, it means 100010001 must have 37 as a factor too.
Lot of weird patterns to be found in here, for sure.
Wow, that's one slightly complex sentence. I did a double-take at first, then had to scan the sentence a second time, to figure out what it meant, because of the consecutive, multiple clauses in it.
Such a number of puns not intended originally, but noticed and italicized ;)
Okay, will confess, I addedone or two puns after the fact, to enhance the result, like the "one" and "complex" at the top, and the words in this sentence.
As the other commenter pointed out, it's just multiples of 111. You can find all sorts of patterns playing around with a calculator, I particularly like this one:
Also, while fooling around with my calculator in school, years ago, I had noticed this pattern:
37 x 3 = 111
37 x 6 = 222
37 x 9 = 333
and so on.
I tried it again just now, all the way up to 37 x 51.
It works perfectly upto 37 x 27 = 999
After that the pattern changes somewhat, but still seems to have some sort of regularity to it.