Define the juggler sequence for a positive integer 
as 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.
Rather surprisingly, all integers appear to eventually reach 1, a conjecture that holds at least up to 
(E. W. Weisstein, 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 A095908),
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 
, 2, ... are 1, 2, 4, 16, 7, 5, 3, 9, 33, 19, 81, 25, 353,
... (OEIS A094670).
