Juggler Sequence
المؤلف:
Sloane, N. J. A.
المصدر:
Sequences A007320, A094679, A095908, and A094670 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
29-10-2020
1022
Juggler Sequence
Define the juggler sequence for a positive integer 
as the sequence of numbers produced by the iteration
![a_(k+1)=<span style=]() {|_a_k^(1/2)_| for even a_k; |_a_k^(3/2)_| for odd a_k, " src="https://mathworld.wolfram.com/images/equations/JugglerSequence/NumberedEquation1.gif" style="height:54px; width:162px" /> |
(1)
|
where 
denotes the floor function. For example, the sequence produced starting with the number 77 is 77, 675, 17537, 2322378, 1523, 59436, 243, 3787, 233046, 482, 21, 96, 9, 27, 140, 11, 36, 6, 2, 1.
Rather surprisingly, all integers appear to eventually reach 1, a conjecture that holds at least up to 
(E. W. Weisstein, Jan. 23, 2006). The numbers of steps 
needed to reach 1 for starting values of 
, 2, ... are 0, 1, 6, 2, 5, 2, 4, 2, 7, 7, 4, 7, 4, 7, 6, 3, 4, 3, 9, 3, ... (OEIS A007320), plotted above. The high-water marks for numbers of steps are 0, 1, 6, 7, 9, 11, 17, 19, 43, 73, 75, 80, 88, 96, 107, 131, ... (OEIS A095908), which occur for starting values of 1, 2, 3, 9, 19, 25, 37, 77, 163, 193, 1119, ... (OEIS A094679).
The smallest integers requiring 
steps to reach 1 for 
, 2, ... are 1, 2, 4, 16, 7, 5, 3, 9, 33, 19, 81, 25, 353, ... (OEIS A094670).
REFERENCES:
Pickover, C. A. Computers and the Imagination. New York: St. Martin's Press, p. 232, 1991.
Pickover, C. A. "Juggler Numbers." Ch. 45 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 102-106 and 301-304, 2002.
Sloane, N. J. A. Sequences A007320, A094679, A095908, and A094670 in "The On-Line Encyclopedia of Integer Sequences."
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