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

[Christian G.Bower]

I did a little calculating here and have some results different from
Singmaster, Bennett and Dunn.

------ Original Message ------
From: Richard Guy <rkg at cpsc.ucalgary.ca>
To: hv at crypt.orgCc: seqfan at ext.jussieu.fr
Subject: Re: n such that there is one sequence of length n having equal sum
and product 

...

> 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
> $N(k)\rightarrow\infty$.

 From the 1998 post I have N(27744)=2 which is higher than their 6324.
I just did a search and found 50 cases of N(k)=2. The largest ten are:
966 1102 2400 2820 4350 4354 5274 6174 6324 27744

I have only 75 cases for N(k)=3, the largest ten:
2520 2590 2964 3024 3195 3450 3954 6906 7220 11874

I have 204 cases for N(k)=4, the largest ten:
12660 13926 14250 16932 20019 21234 22752 28560 29760 47644

Christian