Going for the juggler

It is conjectured that all juggler sequences eventually reach 1. More info: https://t.co/mZOlQgz6KB pic.twitter.com/oo7FxGz1tv

— Cliff Pickover (@pickover) December 5, 2017

Cliff's twitter link ends up here. Eric Weisstein's MathWorld entry is here. Clifford A. Pickover first wrote an article about juggler sequences in November 1990 and challenged its readers to prove that all such sequences fall to 1. He fleshed things out a bit in chapter 40 of his 1991 "Computers and the Imagination". The juggler sequence starting with 37 is:

 0               37

 1              225

 2             3375

 3           196069

 4         86818724

 5             9317

 6           899319

 7        852846071

 8   24906114455136

 9          4990602

10             2233

11           105519

12         34276462

13             5854

14               76

15                8

16                2

17                1

Odd numbers grow the next term; even numbers shrink it. With 37 as our start, it takes 17 steps to get to 1; the largest term (composed of 14 decimal digits) appears here at step 8.

Harry James Smith (27 Jan 1932 - 5 Jun 2010) was an early adherent of the cause and soon found himself looking for record large terms in juggler sequences. In 2008 he found an 89981517-digit term in the sequence starting with 7110201. It's in this spirit that I decided to look for records in either the number of steps needed for a juggler sequence to reach 1 (A007320) or the largest value encountered in a juggler sequence (A094716). Here they are:

# 0           1     0     0            1

# 1           2     1     0            1

# 2           3     6     3            2

# 3           9     7     2            3

# 4          19     9     4            3

# 5          25    11     3            5

# 6          37    17     8           14

# 7          77    19     3            7

# 8         113    16     9           27

# 9         163    43     6           26

#10         173    32    17           82

#11         193    73    47          271

#12        1119    75    49          271

#13        1155    80    24          213

#14        2183    72    32         5929

#15        4065    88    63          386

#16        4229    96    41          114

#17        4649   107    74         1255

#18        7847   131    63         3743

#19       11229   101    54         8201

#20       13325   166    90         1272

#21       15065    66    25        11723

#22       15845   139    43        23889

#23       30817    93    39        45391

#24       34175   193    61         5809

#25       48443   157    60       972463

#26       59739   201    69         5809

#27       78901   258   109       371747

#28      275485   225   148      1909410

#29      636731   263   114       371747

#30     1122603   268   145       209735

#31     1267909   151    99      1952329

#32     1301535   271   122       371747

#33     2263913   298   149       371747

#34     2264915   149    89      2855584

#35     5812827   135    67      7996276

#36     5947165   335   108      3085503

#37     7110201   205   119     89981517

#38    56261531   254    92    105780485

#39    72511173   340   166       621456

#40    78641579   443   275      7222584

#41    92502777   191   117    139486096

#42   125121851   479   203       146173

#43   172376627   262    90    449669621
#44   198424189   484   350      5342028
#45   604398963   327   172    640556693
#46   839327145   224   118   2109464216
#47  1247677915   221   119   3225243807

Five columns: The first is just an identification number. The second column is our starting number. The third column gives the number of steps for that starting number to reach 1. The fourth column gives at which step the starting number reaches a maximum. The fifth column gives the number of decimal digits in that maximum. So you can verify #6 with the example previously provided and note that #37 is as far as Harry J. Smith managed to get. Records are indicated in bold.

To bring the large numbers in a juggler sequence down to a manageable size, one can do a log of the numbers and then a log again. Employing this technique, here's a nice graphic plot of juggler sequences #42 and #43: