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

%I A094679 #26 Apr 15 2025 08:25:21
%S A094679 1,2,3,9,19,25,37,77,163,193,1119,1155,4065,4229,4649,7847,13325,
%T A094679 34175,59739,78901,636731,1122603,1301535,2263913,5947165,72511173,
%U A094679 78641579,125121851,198424189,4488817391
%N A094679 n sets a new record for number of iterations to reach 1 in the juggler sequence problem.
%C A094679 Where records occur in A007320.
%C A094679 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
%C A094679 78901 reaches a maximum of 4064983429...(skip the next 371727 digits)...2140697134 during its trip to 1. - _Robert G. Wilson v_, Jun 14 2004
%C A094679 I postulate that 2 is the only even number in this sequence. - _Harry J. Smith_, Aug 15 2008
%C A094679 a(30) > 1.6*10^9. - _Giovanni Resta_, Apr 08 2017
%H A094679 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JugglerSequence.html">Juggler Sequence</a>
%e A094679 78901 takes 258 iterations to reach 1; see A094698 for the others.
%t A094679 $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_ *)
%Y A094679 Cf. A007320, A094670, A094698, A095906, A094698.
%K A094679 more,nonn,hard
%O A094679 1,2
%A A094679 _Jason Earls_, Jun 09 2004
%E A094679 More terms from _Robert G. Wilson v_, Jun 14 2004
%E A094679 a(25) = 5947165 from _Eric W. Weisstein_, Jan 25 2006
%E A094679 a(26)-a(27) from _Robert G. Wilson v_, Jun 15 2014
%E A094679 a(28)-a(29) from _Giovanni Resta_, Apr 08 2017
%E A094679 a(30) from _Ethan Slota_, Apr 15 2025