wild numbers and feral sequences
Marc LeBrun
mlb at well.com
Fri Jan 19 17:39:46 CET 2001
>=N. J. A. Sloane
> > The following is an attempt at producing a sequence
> > which behaves like the Wild Numbers described by Enoch Haga
> > in his recent posting. The idea is to find a map f from
> > rationals to rationals that occasionally produces an integer.
[...]
> > Obviously many similar examples could be constructed.
Possible candidates might include:
1. Initialize with some a(n) / b(n), then subject the denominators to a
process that chaotically converges to 1, such as the Collatz 3x+1
map. While this is going on do as you please with the numerators to get
the desired kinds of integers at the end.
(Of course embedding Collatz like this would entail an actual open research
problem!)
2. Build something out of Conway's FracTran.
(To recap for the unacquainted: FracTran cleverly views a rational number
as the state of a machine whose "registers" are the exponents of the prime
factorization. Instructions are coded as rationals to multiply this
machine state by to get the next state. Conditionals can be based on the
fact that negative register values imply non-integer state numbers).
3. Use the fact that the continued fraction for most numbers typically
dodders around small values, occasionally spitting out a large term (eg the
292 in pi). For example you might use cbrt(n), perturbing the orbit just
enough to cause a collision with some closely-encountered rational.
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