The juggler sequence for a positive integer 
is the sequence
of numbers produced by the iteration
 |
(1)
|
where 
denotes the floor function. For example, the sequence produced starting with the number 77 is 77, 675,
17537, 2322378, 1523, 59436, 243, 3787, 233046, 482, 21, 96, 9, 27, 140, 11, 36,
6, 2, 1.
All positive integers appear to eventually reach 1, a conjecture that holds at least up to 
(E. W. Weisstein, pers. comm., Jan. 23,
2006). The numbers of steps 
needed to reach 1 for starting values of 
, 2, ... are 0, 1, 6, 2, 5, 2, 4, 2, 7, 7, 4, 7, 4,
7, 6, 3, 4, 3, 9, 3, ... (OEIS A007320), plotted
above. The high-water marks for numbers of steps
are 0, 1, 6, 7, 9, 11, 17, 19, 43, 73, 75, 80, 88, 96, 107, 131, ... (OEIS A094698),
which occur for starting values of 1, 2, 3, 9, 19, 25, 37, 77, 163, 193, 1119, ...
(OEIS A094679).
The smallest integers requiring 
steps to reach 1 for 
, 1, 2, ... are 1, 2, 4, 16,
7, 5, 3, 9, 33, 19, 81, 25, 353, ... (OEIS A094670).
