Proposed sequence

[David Wilson]

I know that Neil doesn't like to include sequences that are not proved, but 
in this
case I hope he would make an exception.

To my knowledge, for n = 3 through 7, the smallest known integers that are a 
sum
of two nth powers of positive rationals but not of two nth powers of 
positive integers
are:

a(3) = 6 = (17/21)^3 + (37/21)^3
a(4) = 5906 = (25/17)^4 + (149/17)^4
a(5) = 68101 = (15/2)^5 + (17/2)^5
a(6) = 1124326946 = (73/5)^6 + (161/5)^6
a(7) = 69071941639 = (63/2)^7 + (65/2)^7

I computed these values about the year 2000, in an attempt to improve one of
Steve Finch's MathSoft pages, which is now located at

http://www.mathsoft.com/mathsoft_resources/unsolved_problems/2186.asp

a(3) and a(4) were already known, and were on Finch's page before I started
work on this problem.  Finch's page indicates that a(3) = 6 has been proved.
a(4) = 5906 was proved in

A. Bremner and P. Morton, A new characterization of the integer 5906,
Manuscripta Math. 44 (1983) 187-229; MR 84i:10016.

This paper also included a guess at a(5) which Finch used on his page when I
first happened upon it, but my value a(5) = 68101 was smaller, and Finch
updated his page when I notified him of it. 68101 is also mentioned in the
same context in the following message dated 9/6/2000:

http://www.math.niu.edu/~rusin/known-math/00_incoming/flt_rings

which says

> Another generalisation is x^k + y^k = n.z^k; which also has not yet
> been solved in the integers.  The example I gave uses k = 5 and
> n = 68101.  It is known that 68101 is the smallest n with k = 5 such
> that there are solutions (and only one primitive solution)....

which idicates that a(5) = 68101 is proved, however, no reference is given.
This note means I am probably not the first discoverer of a(5) = 68101.

However, as far as I can tell, I was the discoverer of a(6) and a(7).  These 
values
are the best known, however, they have not been proved minimal.

The reason I wanted to add this sequence to the OEIS is that Finch's WebSoft
page seems to be the only place on the Web where a(6) and a(7) are recorded,
and I do not want the values lost if the site or page is pulled.

--------------------------------
- David Wilson