"Keith F. Lynch" <kfl@KeithLynch.net> wrote:

For "random" irrational numbers, the expected distribution of terms in distributed fractions ...

Sorry, I meant in *continued* fractions.

My impression from looking at a few of them is that each term is roughly the square of the previous. That raises the question of whether ... any number has terms that consistently grow more slowly, e.g. each term is on average only the 1.5 power of the previous, forever.

That was stupid of me. The difference between 1/(n-1) and 1/n will always be 1/((n-1)*n), so the greedy algorithm will always grab 1/(n-1) rather tnan 1/n if the remainder of the latter would be less than 1/((n-1)*n). The only exception is if it grabbed both, which can only happen as part of a run of consecutive numbers at the beginning of the sequence. So 1/2 + 1/4 + 1/16 + 1/256 + 1/65536 + ... is not a greedy sequence. Still, I wonder whether its sum has any significance. Similarly with other starting points, e.g. 1/3 + 1/9 + 1/81 + 1/6561 ...