# x->x*ceiling(x)

Marc LeBrun mlb at well.com
Mon Sep 2 04:45:59 CEST 2002

It would seem that a natural question might be: How well does this
iteration model a (pseudo) random process?

If it's significantly nonrandom it would help motivate looking deeper for
underlying structure.

As a first cut, for denominator n, the probability of "hitting" a multiple
of n "at random" should (approximately) determine this.

Don't these runs look a little too short?

Of course, since x[n] divides x[n+1], then if n has a lot of divisors the
process can chew them up incrementally instead of having to get lucky all
in one shot, so this makes calculating the expected run length interesting.

Do these numbers match the expected statistics, modified to appropriately
reflect the factorization of n?

Note that if a larger population than the samples here is wanted, the same
clever technology could just as well be applied to compute (n+k)/n for k>1.

(Also, I'm confused: has it been definitely decided that the Pisot
sequences are a red herring, or do I need to dig out the references?)

========
Roland Bacher:

>I cannot prove that the sequence gotten by iterating
>x->x*ceil(x) and starting with x_0=1+1/n   gets always integral,
>but it does so for n=1,....,100.
>
>the first integral indices are:
>
>0, 1, 2, 3, 18, 2, 3, 4, 6, 7, 26, 4, 9, 3, 4, 8, 6, 4, 56, 11, 3, 4, 42,
>4, 33, 7, 5, 4, 38, 5, 79, 6, 4, 15, 14, 8, 200, 29, 13, 5, 36, 3, 4, 5,
>7, 10, 11, 8, 6, 20, 47, 27, 43, 9, 41, 9, 10, 23, 37, 17, 18, 6, 7, 6,
>32, 15, 225, 7, 73, 11, 20, 12, 182, 9, 16, 7, 10, 15, 196, 8, 11, 62, 23,
>5, 26, 4, 5, 8, 85, 11, 18, 61, 22, 177, 59, 10, 187, 10, 20, 6
>
>local maxima are achieved for n=1,2,3,4,5 (18), 11 (26), 19 (56), 37 (200) and
>67 (225)  .
>
>hence x_{18} \in N for x_0=1+1/5;
>
>I have computed these values by doing all computations over the integers
>(multiply by n) and by truncating modulo n^250. This avoids the explosion
>of the integers (of order 2^(2^k) after $k$ iterations)
>and gives the correct answer if the final index i(n) is < 250
>(or perhaps 249 or 248). If the algorithm does not stop befor 245 you should
>increase precision (work with n^500 or even higher).
>
>Roland Bacher