analytics

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...


No comments:

Post a Comment