Definition:Juggler Sequence

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Theorem

Let $m \in \Z_{\ge 0}$ be a positive integer.

The juggler sequence on $m$ is defined recursively as:

$J_m \left({n}\right) = \begin{cases} m & : n = 0 \\ \left\lfloor{\sqrt {J_m \left({n - 1}\right)} }\right\rfloor & : n \text{ even} \\ \left\lfloor{\sqrt {\left({J_m \left({n - 1}\right)}\right)^3} }\right\rfloor & : n \text{ odd} \end{cases}$

where:

$\left\lfloor{x}\right\rfloor$ denotes the floor of $x$
$\sqrt x$ denotes the positive square root of $x$.


Examples

Juggler Sequence on $37$

The Juggler sequence on $37$ is:

$37, 225, 3375, 196069, 86818724, 9317, 899319, 852846071, 24906114455136, 4990602, 2233, 105519, 34276462, 5854, 76, 8, 2, 1$

This sequence is A284987 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Sources