# Joyce Sequence

Antreas P. Hatzipolakis xpolakis at otenet.gr
Mon May 8 08:57:43 CEST 2000

```Michael Kleber wrote:

>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

First, Thanks to Michael for computing the terms,
Second, Neil has added the sequence in EIS (A054382).
Third, Should Neil add the sequences, whose the terms are the numbers
of digits of the (terms of the) sequences n^n, n^n^n^n, etc, and
the like (n!, etc)? My answer is no. IMHO, a sequence has place in EIS if
appears in the literature (mathematical or not); not just because it has
a simple or nice formula.
Fourth, why not "Laisant sequence", since L. had computed the number of digits
of 9^9^9 (in 1906; before Joyce's _Ulysses_)? Well... let's honor with that
name (Joyce sequence) one of the greatest writers ever lived!

Antreas

```