Juggler sequences were publicised by American mathematician and author Clifford A.
[1] The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler.
It is conjectured that all juggler sequences eventually reach 1.
This conjecture has been verified for initial terms up to 106,[3] but has not been proved.
Juggler sequences therefore present a problem that is similar to the Collatz conjecture, about which Paul Erdős stated that "mathematics is not yet ready for such problems".
For example, the juggler sequence starting at a0 = 37 reaches a maximum value of 24906114455136.
Harry J. Smith has determined that the juggler sequence starting at a0 = 48443 reaches a maximum value at a60 with 972,463 digits, before reaching 1 at a157.