On Wed, May 8, 2019 at 5:44 PM <rcs@xmission.com> wrote:
Keith's final example, the numbers 2^X * 3^Y, is relatively easy to generate in sorted order... It looks like the number of active or potentially active ratios is some small number like 4 or 6, and the new-hire and retirement rules should be easy to figure out.
This is a nice idea for analysis, but some of your assertions seem doubtable. Rather than iterating a fractional sequence r_n, we can go through a triple of integers v_n={i,j,k} such that i=0 implies r_n = 2^j/3^k and i=1 implies r_n = 3^j/2^k. At a glance, the iterator v_n -> v_{n+1} does not seem "easy to figure out". Setting aside the boolean sequence for i (?), exponents j & k potentially increase unbounded. These exponent sequences may draw from a finite set locally, but what is that set? Up to 10K I calculated that Numerators: {1, 2, 5, 7, 8, 12, 27, 46, 53, 65, 149} Denominators: {1, 3, 5, 11, 17, 19, 29, 41, 84, 94} and observed that j(n) and k(n) appear to grow very roughly like log(n) as the sequence of r_n approaches 1 asymptotically. After only this, I can not leave off optimistic on the question of finding a term-to-term iterator for the suggested fractional sequence. Maybe it is possible to make better progress starting with the parity sequence: 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0 . . . -- Brad