The fact you have to lay it out like you did kind of points towards the pattern ...

jameshart · 2024-03-29T13:26:23

It’s neat that every third multiple of 37 forms a pattern. It’s just not significant.

Every 271st multiple of 41 also forms a nice pattern:

But that somehow doesn’t seem as interesting.

jameshart · 2024-03-29T14:41:33

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’).

To be concrete:

   111111111111 (12 1s)
   = 111 * 1001001001 (3 1s, 4 1s)
   = 111 * 1001 * 1000001 (3 1s, 2 1s, 2 1s)

   Or

   = 1111 * 100010001 (4 1s, 3 1s)
   = 11 * 101 * 100010001 (2 1s, 2 1s, 3 1s)

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.

fuzztester · 2024-03-29T22:47:29

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 added one or two puns after the fact, to enhance the result, like the "one" and "complex" at the top, and the words in this sentence.