Some time ago, Kellogg communicated to Carmichael a result with an incomplete proof, which was soon after verified as correct. I do not recall the source but recall the result. Define


$$S\_n = \{ (x\_1,x\_2 \dots x\_n) \subset \mathbb{N}^n: \sum x\_i^{-1} = 1\}.$$


And regard elements of $S\_n$ identical up to a permutation as the same element. I am interested in the question: 



> 
> For each $n$, what is the largest integer $l(n)$ such that it appears as a member of some element of $S\_n$?
> 
> 
> 


Define the sequence $u\_n$ with $u\_1 = 1$ and $u\_i = u\_{i-1}(u\_{i-1}+1).$ Then Kellogg reasoned that $l(n)$ is bounded above by $u\_n$.


**The question of this post:** As this was a very old result, has there since arisen a better description of $l(n)$? Exact values for certain arguments, lower bounds, sharper upper bounds?



Maximum in solution set to a Diophantine equation related to unit fractions
$$1={1\over2}+{1\over2}={1\over2}+{1\over3}+{1\over6}={1\over2}+{1\over3}+{1\over7}+{1\over42}={1\over2}+{1\over3}+{1\over7}+{1\over43}+{1\over1806}=\cdots$$ so certainly the sequence $u\_n$, which is $1,2,6,42,1806,\dots$, is a lower bound as well for $l(n)$. 


$u\_n$ is [tabulated](http://oeis.org/A007018) at the Online Encyclopedia of Integer Sequences. 


A proof is attributed to D. R. Curtiss, On Kellogg's Diophantine problem, Amer. Math. Monthly 29 (1922), pp. 380-387. 


Also, the OEIS entry says, 


Using the methods of Aho and Sloane, Fibonacci Quarterly 11 (1973), 429-437, it is easy to show that $a(n)$ is the integer just a tiny bit below the real number $\theta^{2^n}-1/2$, where $\theta \approx 1.597910218$ is the exponential of the rapidly convergent series $\sum\_{n\ge0} \log(1+1/a\_n)/2^{n+1}$. For example, $\theta^{32} - 1/2 \approx 3263442.0000000383$.


Others who have written on this problem are  

Tanzo Takenouchi, On an indeterminate equation, Proc. Phys.-Math. Soc. Japan(3), 3(1921) 78-92.  

David Eppstein, Ten algorithms for Egyptian fractions, Mathematica. in Education and Research, 4(1995) 5-15. This article is also available on the web at <http://www.ics.uci.edu/~eppstein/numth/egypt/intro.html> – see also <https://www.ics.uci.edu/~eppstein/numth/egypt/curtiss.html>  

O. T. Izhboldin & L. D. Kurlyandchik, Unit fractions, Proc. St. Petersburg Math. Soc., 111(1995) 193-200; MR 99m:ll024.


