Juggler sequence

Juggler sequences were publicised by American mathematician and author Clifford A. Pickover.{{cite book |last=Pickover |first=Clifford A. |authorlink=Clifford A. Pickover |date=1992 |title=Computers and the Imagination |publisher=St. Martin's Press |chapter=Chapter 40 |isbn=978-0-312-08343-4 |url-access=registration |url=https://archive.org/details/computersimagina00clif }} 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 |date=2002 |title=The Mathematics of Oz: Mental Gymnastics from Beyond the Edge |url=https://archive.org/details/mathematicsofozm0000pick |url-access=registration |publisher=Cambridge University Press |chapter=Chapter 45: Juggler Numbers |pages=[https://archive.org/details/mathematicsofozm0000pick/page/102 102–106] |isbn=978-0-521-01678-0}}

For example, the juggler sequence starting with a₀ = 3 is

:$a\_1=\; \backslash lfloor\; 3^\backslash frac\{3\}\{2\}\; \backslash rfloor\; =\; \backslash lfloor\; 5.196\backslash dots\; \backslash rfloor\; =\; 5,$

:$a\_2=\; \backslash lfloor\; 5^\backslash frac\{3\}\{2\}\; \backslash rfloor\; =\; \backslash lfloor\; 11.180\backslash dots\; \backslash rfloor\; =\; 11,$

:$a\_3=\; \backslash lfloor\; 11^\backslash frac\{3\}\{2\}\; \backslash rfloor\; =\; \backslash lfloor\; 36.482\backslash dots\; \backslash rfloor\; =\; 36,$

:$a\_4=\; \backslash lfloor\; 36^\backslash frac\{1\}\{2\}\; \backslash rfloor\; =\; \backslash lfloor\; 6\; \backslash rfloor\; =\; 6,$

:$a\_5=\; \backslash lfloor\; 6^\backslash frac\{1\}\{2\}\; \backslash rfloor\; =\; \backslash lfloor\; 2.449\backslash dots\; \backslash rfloor\; =\; 2,$

:$a\_6=\; \backslash lfloor\; 2^\backslash frac\{1\}\{2\}\; \backslash rfloor\; =\; \backslash lfloor\; 1.414\backslash dots\; \backslash rfloor\; =\; 1.$

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 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, about which Paul Erdős stated that "mathematics is not yet ready for such problems".

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:

:

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.[https://web.archive.org/web/20091027155431/http://geocities.com/hjsmithh/Juggler/Juggle3L.html Letter from Harry J. Smith to Clifford A. Pickover, 27 June 1992]

• Arithmetic dynamics
• Collatz conjecture
• Recurrence relation

• {{Mathworld|id=JugglerSequence}}
• Juggler sequence (A094683) at the On-Line Encyclopedia of Integer Sequences. See also:
• Number of steps needed for juggler sequence (A094683) started at n to reach 1.
• n sets a new record for number of iterations to reach 1 in the juggler sequence problem.
• Number of steps where the Juggler sequence reaches a new record.
• Smallest number which requires n iterations to reach 1 in the juggler sequence problem.
• Starting values that produce a larger juggler number than smaller starting values.
• [https://web.archive.org/web/20110607051447/http://members.chello.nl/k.ijntema/juggler.html Juggler sequence calculator] at Collatz Conjecture Calculation Center
• [https://web.archive.org/web/20091027103635/http://geocities.com/hjsmithh/Juggler/index.html Juggler Number pages] by Harry J. Smith

Category:Arithmetic dynamics

Category:Integer sequences

Category:Recurrence relations

Category:Unsolved problems in number theory