Juggler sequence
# Juggler sequence

In number theory, a juggler sequence is an integer sequence that starts with a positive integer $a_0,$ with each subsequent term in the sequence defined by the recurrence relation:

$a_{k+1}={\begin{cases}\left\lfloor a_{k}^{\frac {1}{2}}\right\rfloor ,&{\text{if }}a_{k}{\text{ is even}}\\\\\left\lfloor a_{k}^{\frac {3}{2}}\right\rfloor ,&{\text{if }}a_{k}{\text{ is odd}}.\end{cases}}$

## Background

Juggler sequences were publicised by American mathematician and author Clifford A. Pickover. The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler.

For example, the juggler sequence starting with $a_0$ = 3 is

    $a_{1}=\lfloor 3^{\frac {3}{2}}\rfloor =\lfloor 5.196\dots \rfloor =5,$
    $a_{2}=\lfloor 5^{\frac {3}{2}}\rfloor =\lfloor 11.180\dots \rfloor =11,$
    $a_{3}=\lfloor 11^{\frac {3}{2}}\rfloor =\lfloor 36.482\dots \rfloor =36,$
    $a_{4}=\lfloor 36^{\frac {1}{2}}\rfloor =\lfloor 6\rfloor =6,$
    $a_{5}=\lfloor 6^{\frac {1}{2}}\rfloor =\lfloor 2.449\dots \rfloor =2,$
    $a_{6}=\lfloor 2^{\frac {1}{2}}\rfloor =\lfloor 1.414\dots \rfloor =1.$

If a juggler sequence reaches 1, then all subsequent terms are equal to 1. It is conjectured that all juggler sequences eventually reach 1. This conjecture has been verified for initial terms up to $10^6,$ 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 a given initial term n, one defines l(n) to be the number of steps which the juggler sequence starting at n takes to first reach 1, and h(n) to be the maximum value in the juggler sequence starting at n. For small values of n we have:

    n Juggler sequence l(n) (sequence A007320 in the OEIS) h(n) (sequence A094716 in the OEIS) 2 2, 1 1 2 3 3, 5, 11, 36, 6, 2, 1 6 36 4 4, 2, 1 2 4 5 5, 11, 36, 6, 2, 1 5 36 6 6, 2, 1 2 6 7 7, 18, 4, 2, 1 4 18 8 8, 2, 1 2 8 9 9, 27, 140, 11, 36, 6, 2, 1 7 140 10 10, 3, 5, 11, 36, 6, 2, 1 7 36

Juggler sequences can reach very large values before descending to 1. For example, the juggler sequence starting at $a_0$ = 37 reaches a maximum value of 24906114455136. Harry J. Smith has determined that the juggler sequence starting at $a_0$ = 48443 reaches a maximum value at $a_{60}$ with 972,463 digits, before reaching 1 at $a_{157}.$

