Main Concept

A juggler sequence is an integer sequence which begins with a positive integer $\mathrm{a\_\_0}$ and has each subsequent term defined by the recursive relation:

 $a_k$$\=\left\{\begin{array}{cc}⌊a_k^{\frac{1}{2}}⌋& \mathbf{if} a_k \mathrm{is} \mathrm{even}\\ ⌊a_k^{\frac{3}{2}}⌋& \mathbf{if} a_k \mathrm{is} \mathrm{odd}\end{array}\right.$  , where $⌊x⌋$ denotes the floor function.

The name of this sequence comes from the rising and falling nature of the terms, which resembles the movement of balls or clubs in the hands of a juggler. After a juggler sequence reaches 1, all subsequent terms equal 1. It is conjectured that all juggler sequences eventually descend to 1, no matter what positive integer they begin with.

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