Define the juggler sequence for a positive integer  as the sequence of numbers produced by the iteration

{|_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."