Juggler sequence

In recreational mathematics 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:

:

Juggler sequences were publicised by American mathematician and author Clifford A. Pickover. [cite book | last = Pickover
first = Clifford A.
authorlink = Clifford A. Pickover
title = Computers and the Imagination
publisher = St. Martin's Press
date = 1992
pages = Chapter 40
isbn = 978-0312083434] The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler. [cite book
last = Pickover
first = Clifford A.
authorlink = Clifford A. Pickover
title = The Mathematics of Oz
publisher = Cambridge University Press
date = 2002
pages = Chapter 45
isbn = 978-0521016780]

For example, the juggler sequence starting with "a"₀ = 3 is

:$a\_1=\; lfloor\; 3^frac\{3\}\{2\}\; floor\; =\; lfloor\; 5.196dots\; floor\; =\; 5,$:$a\_2=\; lfloor\; 5^frac\{3\}\{2\}\; floor\; =\; lfloor\; 11.180dots\; floor\; =\; 11,$:$a\_3=\; lfloor\; 11^frac\{3\}\{2\}\; floor\; =\; lfloor\; 36.482dots\; floor\; =\; 36,$:$a\_4=\; lfloor\; 36^frac\{1\}\{2\}\; floor\; =\; lfloor\; 6\; floor\; =\; 6,$:$a\_5=\; lfloor\; 6^frac\{1\}\{2\}\; floor\; =\; lfloor\; 2.449dots\; floor\; =\; 2,$:$a\_6=\; lfloor\; 2^frac\{1\}\{2\}\; floor\; =\; lfloor\; 1.414dots\; floor\; =\; 1.$

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 verifed for initial terms up to 10⁶, [*MathWorld|title= Juggler Sequence|urlname= JugglerSequence] but has not been proved. Juggler sequences therefore present a problem that is similar to the Collatz conjecture.

For a given initial term "n" we define "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:

:

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. [ [http://www.geocities.com/hjsmithh/Juggler/Juggle3L.html Letter from Harry J. Smith to Cliiford A. Pickover, 27th June 1992] ]

References

External links

* [http://members.chello.nl/k.ijntema/juggler.html Juggler sequence calculator] at Collatz Conjecture Calculation Center
* [http://www.geocities.com/hjsmithh/Juggler/index.html Juggler Number pages] by Harry J. Smith