Monday, October 5, 2015

Greedy primorial fractions

I noticed an interesting thing with this sum:
where the # is the primorial, or the product of the primes up to the ith prime...
For example above i=6, that if you add 1 to the numerator of the last term you get exactly 1... Like:
So if the bottom one above was one less than the prime 11 it is just a number very close to 1 like:

So it is in fact a greedy representation of 1 over the primorials or the largest numerators such that the sum is less than 1 as the series continues... The terms get small very quickly...

**Note** It also works with just the regular factorial:

It's interesting to note that the denominators are the same as the general power series... And also the first i terms equal the infinite terms to the right so it can be used as a general numbering system, but one where the "base" of a digit increases as you go to the right...
So separating "digits" by commas you could have a number like:
0, 1, 2, 3 just so the ith digit is less than i which would be:
If I'm thinking write that would be the unique terminating representation of 11/40 in this system but I'm not sure... of course it would be the same number as 0,1,2,3,4,5,6,7,... Of course this is all taking place right of the decimal point, but I think it works to the left too with the fractions upside down with the possible size of the digit increasing to the right and left of the decimal point...

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