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In number theory, a juggler sequence is an integer sequence that starts with a positive integer a₀, 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 publicized by American mathematician and author 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.^[2]

For example, the juggler sequence starting with a₀ = 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 all initial terms up to 7110200, so that 7110201 is the first number that lacks verification, but has not been proven or disproven.^[3]

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 a₀ = 37 reaches a maximum value of 24906114455136. Harry J. Smith has determined that the juggler sequence starting at a₀ = 48443 reaches a maximum value at a₆₀ with 972,463 digits, before reaching 1 at a₁₅₇.^[4]

References

^ Pickover, Clifford A. (1992). “Chapter 40”. Computers and the Imagination. St. Martin’s Press. ISBN 978-0-312-08343-4.
^ Pickover, Clifford A. (2002). “Chapter 45: Juggler Numbers”. The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. Cambridge University Press. pp. 102–106. ISBN 978-0-521-01678-0.
^ derneueschwan.ch/juggler, 12 May 2026
^ Letter from Harry J. Smith to Clifford A. Pickover, 27 June 1992

External links

Weisstein, Eric W. “Juggler sequence”. MathWorld.
Juggler sequence (A094683) at the On-Line Encyclopedia of Integer Sequences. See also:
Juggler sequence calculator at Collatz Conjecture Calculation Center
Juggler Number pages by Harry J. Smith

[rdp-we-cite_note-1] Pickover, Clifford A. (1992). “Chapter 40”. Computers and the Imagination. St. Martin’s Press. ISBN 978-0-312-08343-4.

[rdp-we-cite_note-2] Pickover, Clifford A. (2002). “Chapter 45: Juggler Numbers”. The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. Cambridge University Press. pp. 102–106. ISBN 978-0-521-01678-0.

[rdp-we-cite_note-3] rneueschwan.ch/juggler, 12 May 2026

[rdp-we-cite_note-4] Letter from Harry J. Smith to Clifford A. Pickover, 27 June 1992

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Sample Page

Background

See also

References

External links