rough notes on finding near misses to cycles in juggler sequence

this is about feeding a number back into a function over and over again, and seeing what kind of patterns we get.

see the entry for the collatz conjecture in the complexity lab manual, it is a similar game

def jug(n) = {int(sqrt(n^3))  if n is odd else  int(sqrt(n))}

int(r) is the greatest integer less than r, i.e. int(4.773)=4

so that jug(2)=2, and jug(2)=1 and jug(1)=1 and we stop there.  that's a fixed point.

jug(3)=int(sqrt(27))=5, and jug(5)=11, etc..

the sequence of numbers we get is called the orbit of the first number under the dynamical system for our function jug(x)

here are the orbits for the first 9 integers
[1]
[2, 1]
[3, 5, 11, 36, 6, 2, 1]
[4, 2, 1]
[5, 11, 36, 6, 2, 1]
[6, 2, 1]
[7, 18, 4, 2, 1]
[8, 2, 1]
[9, 27, 140, 11, 36, 6, 2, 1]
>>>

here is orbit for 37:

 [37, 225, 3375, 196069, 86818724, 9317, 899319, 852846071, 24906114455136, 4990602, 2233, 105519, 34276462, 5854, 76, 8, 2, 1]

it seems that no matter what n we start off with we reach 1 (i think this has been tested up to a million)

no one knows how to prove that this is so.  There are many systems like this and they puzzle us.
wiki on the juggler sequence

the question is: are there numbers for which the orbit reaches a fixed point?  or are there numbers for which the orbit cycles?  for instance, look at the orbit for 3:
[3, 5, 11, 36, 6, 2, 1]
if the square root of 36 had happened to be 5 this orbit would have been:
3, 5, 11, 36, 5, 11, 36, 5...

and 5,11,36 would be a cycle.  no one has found any cycles yet, but lets look for almost cycles like 3, 5, 11, 36, 6... 6 is CLOSE to 5.


so,

i generate an orbit under this dynamical system.  [(x0,0), (x1,1)....(xn,n)]
then i sort this by x, so that xi that are near each other in size are near each other in the sequence

then i create a sequence of the differences between consecutive xi's  and sort that.  here is what i get for 37:

(diff, (xi, i), (xj, j))
(1L, (1L, 17), (2L, 16))
(6L, (2L, 16), (8L, 15))
(9L, (8L, 15), (17, 18))
(20, (17, 18), (37, 0))
(39L, (37, 0), (76L, 14))
(149L, (76L, 14), (225, 1))
(1142L, (2233L, 10), (3375, 2))
(2008L, (225, 1), (2233L, 10))
(2479L, (3375, 2), (5854L, 13))
(3463L, (5854L, 13), (9317, 5))
(90550L, (105519L, 11), (196069, 3))
(96202L, (9317, 5), (105519L, 11))
(703250, (196069, 3), (899319, 6))
(4091283L, (899319, 6), (4990602L, 9))
(29285860L, (4990602L, 9), (34276462L, 12))
(52542262L, (34276462L, 12), (86818724, 4))
(766027347, (86818724, 4), (852846071, 7))
(24905261609065L, (852846071, 7), (24906114455136L, 8))


now if i want to search for orbits with near misses... what do i search for?  obviously the first few entries do not count

the 4th and 5th entries are close to what i'm looking for.  how do i write an algorithm to pick them out?  the diff must be small but the indexes must be far apart.

the 3rd entry has indices 3 apart but i'm not going to count that as a good candidate!

hmm... a programming puzzle!

this post is continued in the comments...