# Juggler sequence

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Juggler_sequence
> Markdown URL: https://mediated.wiki/source/Juggler_sequence.md
> Source: https://en.wikipedia.org/wiki/Juggler_sequence
> Source revision: 1356754242
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Integer sequence in number theory

Not to be confused with [Juggling pattern](/source/Juggling_pattern).

In [number theory](/source/Number_theory), a **juggler sequence** is an [integer sequence](/source/Integer_sequence) that starts with a [positive integer](/source/Positive_integer) *a*0, with each subsequent term in the sequence defined by the [recurrence relation](/source/Recurrence_relation): a k + 1 = { ⌊ a k 1 2 ⌋ , if a k is even ⌊ a k 3 2 ⌋ , if a k is odd . {\displaystyle 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 publicized by American mathematician and author [Clifford A. Pickover](/source/Clifford_A._Pickover).[1] The name is derived from the rising and falling nature of the sequences, like balls in the hands of a [juggler](/source/Juggler).[2]

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

- a 1 = ⌊ 3 3 2 ⌋ = ⌊ 5.196 … ⌋ = 5 , {\displaystyle a_{1}=\lfloor 3^{\frac {3}{2}}\rfloor =\lfloor 5.196\dots \rfloor =5,}

- a 2 = ⌊ 5 3 2 ⌋ = ⌊ 11.180 … ⌋ = 11 , {\displaystyle a_{2}=\lfloor 5^{\frac {3}{2}}\rfloor =\lfloor 11.180\dots \rfloor =11,}

- a 3 = ⌊ 11 3 2 ⌋ = ⌊ 36.482 … ⌋ = 36 , {\displaystyle a_{3}=\lfloor 11^{\frac {3}{2}}\rfloor =\lfloor 36.482\dots \rfloor =36,}

- a 4 = ⌊ 36 1 2 ⌋ = ⌊ 6 ⌋ = 6 , {\displaystyle a_{4}=\lfloor 36^{\frac {1}{2}}\rfloor =\lfloor 6\rfloor =6,}

- a 5 = ⌊ 6 1 2 ⌋ = ⌊ 2.449 … ⌋ = 2 , {\displaystyle a_{5}=\lfloor 6^{\frac {1}{2}}\rfloor =\lfloor 2.449\dots \rfloor =2,}

- a 6 = ⌊ 2 1 2 ⌋ = ⌊ 1.414 … ⌋ = 1. {\displaystyle 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 all initial terms up to 7110200, so that 7110201 is the first number that lacks verification, but has not been proven or disproven.[3] Juggler sequences therefore[*[clarification needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify)*] present a problem similar to the [Collatz conjecture](/source/Collatz_conjecture), about which [Paul Erdős](/source/Paul_Erd%C5%91s) stated that "mathematics may not be ready for such problems."[4]

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.[5]

## See also

- [Arithmetic dynamics](/source/Arithmetic_dynamics#Other_areas_in_which_number_theory_and_dynamics_interact)

- [Collatz conjecture](/source/Collatz_conjecture)

- [Recurrence relation](/source/Recurrence_relation)

## References

1. **[^](#cite_ref-1)** [Pickover, Clifford A.](/source/Clifford_A._Pickover) (1992). "Chapter 40". [*Computers and the Imagination*](https://archive.org/details/computersimagina00clif). St. Martin's Press. [ISBN](/source/ISBN_(identifier)) [978-0-312-08343-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-312-08343-4).

1. **[^](#cite_ref-2)** [Pickover, Clifford A.](/source/Clifford_A._Pickover) (2002). "Chapter 45: Juggler Numbers". [*The Mathematics of Oz: Mental Gymnastics from Beyond the Edge*](https://archive.org/details/mathematicsofozm0000pick). Cambridge University Press. pp. [102–106](https://archive.org/details/mathematicsofozm0000pick/page/102). [ISBN](/source/ISBN_(identifier)) [978-0-521-01678-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-01678-0).

1. **[^](#cite_ref-3)** [derneueschwan.ch/juggler, 12 May 2026](https://derneueschwan.ch/juggler/)

1. **[^](#cite_ref-4)** Guy, Richard (2004-07-13). [*Unsolved Problems in Number Theory*](https://books.google.com/books?id=1AP2CEGxTkgC&pg=PA330). Springer Science & Business Media. pp. 330–36. [ISBN](/source/ISBN_(identifier)) [978-0-387-20860-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-20860-2).

1. **[^](#cite_ref-5)** [Letter from Harry J. Smith to Clifford A. Pickover, 27 June 1992](https://web.archive.org/web/20091027155431/http://geocities.com/hjsmithh/Juggler/Juggle3L.html)

## External links

- [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Juggler sequence"](https://mathworld.wolfram.com/JugglerSequence.html). *[MathWorld](/source/MathWorld)*.

- [Juggler sequence](https://oeis.org/A094683) (A094683) at the [On-Line Encyclopedia of Integer Sequences](/source/On-Line_Encyclopedia_of_Integer_Sequences). See also: - [Number of steps needed for juggler sequence (A094683) started at n to reach 1.](https://oeis.org/A007320) - [n sets a new record for number of iterations to reach 1 in the juggler sequence problem.](https://oeis.org/A094679) - [Number of steps where the Juggler sequence reaches a new record.](https://oeis.org/A094698) - [Smallest number which requires n iterations to reach 1 in the juggler sequence problem.](https://oeis.org/A094670) - [Starting values that produce a larger juggler number than smaller starting values.](https://oeis.org/A143742)

- [Juggler sequence calculator](https://web.archive.org/web/20110607051447/http://members.chello.nl/k.ijntema/juggler.html) at Collatz Conjecture Calculation Center

- [Juggler Number pages](https://web.archive.org/web/20091027103635/http://geocities.com/hjsmithh/Juggler/index.html) by Harry J. Smith

---
Adapted from the Wikipedia article [Juggler sequence](https://en.wikipedia.org/wiki/Juggler_sequence) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Juggler_sequence?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.