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

[Richard Guy]

No, I don't know anything more about it
than you do.  Here is the present state
of UPINT D24.  Any further information that
you, or anyone else, can give will be most
welcome.     R.

\usection{D24}{Sum equals product.}

For the case $k=3$ with sum and product 1 there is
an extensive literature.  See {\it MR} {\bf99a}:11034
for a pocket survey.

For $k>2$ the equation $a_1a_2\cdots a_k=a_1+a_2+\cdots+a_k$
has the solution $a_1=2$, $a_2=k$, $a_3=a_4=\cdots=a_k=1$.
Schinzel showed that there is no other solution
in positive integers for $k=6$ or $k=24$.
Misiurewicz has shown that $k=2$, 3, 4, 6, 24, 114 (misprinted
as 144 in {\it Elem.\ Math.}\ and in the first edition
of this book), 174 and 444 are the only $k<1000$ for which
there is exactly one solution.  The search has been
extended by Singmaster, Bennett \& Dunn to
$k\le1440000$.   They let $N(k)$ be the
number of different `sum = product' sequences of
size $k$, and conjecture that $N(k)>1$ for
all $k>444$.  They find that $N(k)=2$
for 49 values of $k$ up to 120000, the largest
being 6174 and 6324, and conjecture that $N(k)>2$
for $k>6324$.  They also find that $N(k)=3$
for 78 values of $k$ in the same range, the
largest being 7220 and 11874, and conjecture
that $N(k)>3$ for $k>11874$; also that
$N(k)\rightarrow\infty$.

This problem seems first to have been asked by Trost,
arising from the solution of $a_1a_2\cdots a_k=a_1+a_2+\cdots+a_k=1$ in
rationals. For $k=3$ this is due to Sierpi\'nski; for
$k>3$ to Schinzel.
\hGidx{Sierpi\'nski, Wac{\l}aw} \hGidx{Schinzel, Andrzej}

On 96-08-19, a month before he died, Erd\H{o}s wrote:

\begin{quotation}
  Do you know anything about the equation
$$x_1\cdot x_2\cdot \cdots \cdot x_n=n(x_1+x_2+\cdots+x_n),$$
$x_1\leq x_2\leq\ldots\leq x_n$ where the $x_i$ are
positive integers?  Denote the number of solutions
in $n$ by $f(n)$; $f(n)>n^{\epsilon}$ for some
$\epsilon$ seems to be true?  $n$ is a {\bf champion} if
$f(n)>f(m)$ for all $m<n$, $n$ is an {\bf antichampion}
if $f(m)>f(n)$ for all $m>n$. The antichampions are
perhaps always primes, but this would be the
\Gidx{Law of Small Numbers}.

Let $a_1<a_2<\ldots<a_k\leq x$ be a sequence of
integers $\leq x$ no one of which divides any other.
$\max k=\lfloor(x+1)/2\rfloor$ is of course well
known. Let $A=\{a_1<a_2<\ldots<a_k\leq x\}$; denote
by $f(A;n)$ the number of $a_i\leq n$.  How large
can $\sum_{n=1}^x f(A;n)$ be?  Is it true that the
maximum is $(x^2)/6+O(x)$? The maximum is given
perhaps by the integers $x/3\leq a_i<2x/3$.
\end{quotation}

\medskip

\small

\Aidx{BrownM}
M.~L.~Brown, On the diophantine equation $\sum X_i=\prod X_i$,
{\it Math.\ Comput.}, {\bf42}(1984) 239--240; {\it MR} {\bf85d}:11030.

\Aidx{EckerM}
Michael W.~Ecker, When does a sum of positive integers
equal their product? {\it Math.\ Mag.}, {\bf75}(2002) 41--47.

\Aidx{Garaev}
M.~Z.~Garaev, On the Diophantine equation
$(x+y+z)^3=nxyz$,
{\it Vestnik Moskov.\ Univ.\ Ser.\  I Mat.\ Mekh.},
{\bf2001} 66--67, 72; transl.\ {\it Moscow
Univ.\ Math.\ Bull.}, {\bf56}(2001) 46;
{\it MR} {\bf2002e}:11032.

\Aidx{Misiu}
M.~Misiurewicz, Ungel\"oste Probleme, {\it Elem.\ Math.},
{\bf21}(1966) 90.

\Aidx{Schin}
A.~Schinzel, Triples of positive integers with the
same sum and the same product, {\it Serdica Math.\ L.},
{\bf22}(1996) 587--588; {\it MR} {\bf98g}:11033.

\Aidx{Singm}
David Singmaster, Untitled Brain Twister, {\it The
Daily Telegraph} weekend section, 88-04-09 \& 16.

\Aidx{Benne}\Aidx{Dunn}
David Singmaster, Mike Bennett \& Andrew Dunn,
Sum = product sequences, (July 1988 preprint).

\Aidx{Starke}
E.~P.~Starke \& others, Solution to Problem E2262
[1970, 1008] \& editorial comment,
{\it Amer.\ Math.\ Monthly} {\bf78}(1971) 1021--1022.

\Aidx{Trost}
E.~Trost, Ungel\"oste Probleme, Nr.\ {\bf14},
{\it Elem.\ Math.}, {\bf11}(1956) 134--135;
\& {\bf21}(1966) 90.

\Aidx{Viola}
C.~Viola, On the diophantine equations
$\prod_0^kx_i-\sum_0^kx_i=n$ and
$\sum_0^k\frac{1}{x_i}=\frac{a}{n}$, {\it Acta Arith.},
{\bf22}(1972/3) 339--352; {\it MR} {\bf48} \#234.

\normalsize

On Thu, 7 Apr 2005 hv at crypt.org wrote:

> 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