Juggler sequence

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:

$${\displaystyle a_{k+1}={\begin{cases}\left\lfloor a_{k}^{\frac {1}{2}}\right\rfloor ,&{\text{if }}a_{k}{\text{ is even}}\\\\\left\lfloor a_{k}^{\frac {3}{2}}\right\rfloor ,&{\text{if }}a_{k}{\text{ is odd}}.\end{cases}}}$$

Background

Juggler sequences were publicised 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

${\displaystyle a_{1}=\lfloor 3^{\frac {3}{2}}\rfloor =\lfloor 5.196\dots \rfloor =5,}$

${\displaystyle a_{2}=\lfloor 5^{\frac {3}{2}}\rfloor =\lfloor 11.180\dots \rfloor =11,}$

${\displaystyle a_{3}=\lfloor 11^{\frac {3}{2}}\rfloor =\lfloor 36.482\dots \rfloor =36,}$

${\displaystyle a_{4}=\lfloor 36^{\frac {1}{2}}\rfloor =\lfloor 6\rfloor =6,}$

${\displaystyle a_{5}=\lfloor 6^{\frac {1}{2}}\rfloor =\lfloor 2.449\dots \rfloor =2,}$

${\displaystyle a_{6}=\lfloor 2^{\frac {1}{2}}\rfloor =\lfloor 1.414\dots \rfloor =1.}$

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 initial terms up to 10⁶,^[3] 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:

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]

See also

• Arithmetic dynamics
• Collatz conjecture
• Recurrence relation

References

External links

• Weisstein, Eric W. "Juggler sequence". MathWorld.
• 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.
• Juggler sequence calculator at Collatz Conjecture Calculation Center
• Juggler Number pages by Harry J. Smith