A094679 n sets a new record for number of iterations to reach 1 in the juggler sequence problem.

1, 2, 3, 9, 19, 25, 37, 77, 163, 193, 1119, 1155, 4065, 4229, 4649, 7847, 13325, 34175, 59739, 78901, 636731, 1122603, 1301535, 2263913, 5947165, 72511173, 78641579, 125121851, 198424189, 4488817391

Offset: 1

Author

Jason Earls, Jun 09 2004

Keywords

• more
• nonn
• hard

Comments

Where records occur in A007320.

The Juggler sequence: begin with x and if x is even, [sqrt(x)] -> x and if x is odd, [sqrt(x^3)] -> x and repeat until x = 1, count the iterations. - Robert G. Wilson v, Jun 14 2004

78901 reaches a maximum of 4064983429...(skip the next 371727 digits)...2140697134 during its trip to 1. - Robert G. Wilson v, Jun 14 2004

I postulate that 2 is the only even number in this sequence. - Harry J. Smith, Aug 15 2008

a(30) > 1.6*10^9. - Giovanni Resta, Apr 08 2017

Examples

			78901 takes 258 iterations to reach 1; see A094698 for the others.
		

Links

• Eric Weisstein's World of Mathematics, Juggler Sequence

Crossrefs

Cf. A007320, A094670, A094698, A095906, A094698.

Programs

Mathematica

$MaxPrecision = 250000000; js[n_] := If[ EvenQ[ n], Floor[ Sqrt[n]], Floor[ Sqrt[n^3]]]; f[n_] := Block[{c = 1, k = n}, While[k = js[k]; k != 1, c++ ]; c]; a = {0}; Do[ b = f[n]; If[b > a[[ -1]], AppendTo[a, b]], {n, 3053595}]; a (* Robert G. Wilson v *)

Extensions

More terms from Robert G. Wilson v, Jun 14 2004

a(25) = 5947165 from Eric W. Weisstein, Jan 25 2006

a(26)-a(27) from Robert G. Wilson v, Jun 15 2014

a(28)-a(29) from Giovanni Resta, Apr 08 2017

a(30) from Ethan Slota, Apr 15 2025