n such that there is one sequence of length n having equal sum and product

[hv at crypt.org]

Louis Marmet <louis at marmet.ca> wrote:
:>I am interested in the series of "numbers n such that there is just one 
:>sequence of length n having equal sum and product".  According to the 
:>"On-Line Encyclopedia of Integer sequences", only 8 terms are known {2, 
:>3, 4, 6, 24, 114, 174, 444}.
:>See: 
:>http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A033179
:>A reference to R. K. Guy's book 'Unsolved Problems in Number Theory' 
:>(Section D24) is given and I wonder how far this series has been tested 
:>(I don't have the book).  I can only suppose there is no proof that the 
:>series is finite (or infinite).
:>
:>  I tested it up to 3634884924 but haven't found any other terms to the 
:>series.  I am writing a web page explaining how I searched these numbers 
:>(page not finished yet...and  I am still checking my algorithm too... 
:>http://www.marmet.ca/louis/sumprod/index.html)

Hmm, I was confused about precisely what "sequences" were being considered,
since it seems to me that k=2 is satisfied by both [ 0, 0 ] and [ 2, 2 ],
so I had a look at A033179 which enlightened me not at all. Then I looked
at D24 in Richard Guy's excellent book and found no more information there
either. (I'm looking at the 2nd edition, apologies if this has already
been addressed in the 3rd.)

It doesn't appear to be being treated as a "trivial" solution so I guess
it is somehow considered invalid: are the a_i constrained to be positive
integers? I think the book and the EIS sequence would both benefit from
clarification.

I think it is also worth mentioning the minimal constraint that terms > 2
in A033179 must be of the form p+1 (since [ a+1, b+1, <1> x (ab - 1) ]
is always a solution for k = ab+1, and non-trivial when a >= b >= 2).

Hugo van der Sanden