Joyce Sequence, A054382

Antreas P. Hatzipolakis xpolakis at otenet.gr
Tue May 9 20:25:23 CEST 2000

```>Et al,
>
>In[3]:=Table[Ceiling[Log[10,j]*j^j],{j,1,20}]
>
>Out[3]=
>{0, 2, 13, 155, 2185, 36306, 695975, 15151336, 369693100, 10000000000,
^                                                        ^^^^^^^^^^^^
----------------------------------------------------------------------------

Date: Sun, 7 May 2000 20:37:33 -0400 (EDT)
From: Michael Kleber <kleber at math.mit.edu>
To: kleber at math.mit.edu, xpolakis at otenet.gr
Subject: Re: Joyce Sequence
Cc: math-fun at optima.CS.Arizona.EDU

Oops -- #digits is 1+floor(log()), not ceiling(log()), so my 10000000000
ought to be 10000000001...

--mk

> Quoth Antreas P. Hatzipolakis:
>
> > For a_n = n^n^n, let's define as "Joyce's sequence" the sequence:
> > j_n = the number of digits of a_n, that is:
> > 1, 2, j_3, j_4, j_5, j_6, j_7, j_8, 369693100, j_10, .....
> >
> > This sequernce is of some historical interest (see below).
> >
> > Question: j_3,...,j_8  ??
> ...
> > C. A. Laisant (1906) proved that the number of digits of Joyce's number
> > (9^9^9) is 369,693,100.  H. S. Uhler (1947) published the log of the number
> > to 250 decimal places.
>
> Happily, digital computers have come a long way since 1906.
>
>   % math
>   Mathematica 4.0 for Solaris
>   Copyright 1988-1999 Wolfram Research, Inc.
>    -- Motif graphics initialized --
>
>   In[1]:= f[j_] := Ceiling[ Log[10,j] j^j ]
>
>   In[2]:= Table[ f[j], {j,2,10} ]
>
>   Out[2]= {2, 13, 155, 2185, 36306, 695975, 15151336, 369693100, 10000000000}
>
> --Michael Kleber
>   kleber at math.mit.edu

----------------------------------------------------------------------------

>297121486765, 9622088391635, 337385711567665,
>  12735782555419983, 515003176870815368, 22212093154093428530,
>1017876887958723919835, 49390464231494436119285,
>  2529911258871481925207097, 136422882873335474575244148}
>
>But we know that the first entry should be 1 and not 0.
>
>Sequentially yours,
>Robert G. Wilson V
>rgwv at kspaint.com

APH

```