My algorithm is too slow

Reposted from Stack Overflow - I think this is a more appropriate place to ask the question.

I have an algorithm that for an integer x and a starting integer i such that 1 < i < x the next value of i is computed by i = floor(x / i) + (x mod i). This continues until we reach an i that we've already seen.

In JavaScript (though this question is language agnostic):

function f(x, i) {
    var map = {};
    while(!map[i]) {
        map[i] = true;
        i = Math.floor(x / i) + (x % i); // ~~(x / i) is a faster way of flooring
    }
    return i;
}

I can prove that we will eventually reach an i we've already seen, but I'm wondering:

1. Is there is a more efficient way of computing the next i?
2. (More importantly) Is there is a way to compute the nth i without running through the loop n times?

Just to clarify - I know there are faster ways than using JS hash maps for that check, and that flooring can be replaced by integer division in other languages. I have made both of those optimizations, but I left them out to try to make the code easier to understand.