[seqfan] Is the set of numbers representable by sum of 5 positive cubes in exactly 3 ways finite?

[Sean A. Irvine]

Hi all,

Dave Consiglio has been working to fill some holes in the OEIS around
sequences arising from sums of fixed numbers of like powers.  There are
some interesting cases arising, but I will mention just one sequence here.
A343705 is the set of numbers that are the sum of five positive cubes in
exactly three ways:

https://oeis.org/A343705

Could this sequence be finite?

Generating terms up to a(18984) = 2715466 is fairly easy (the b-file), but
thereafter we find no further terms with our simple search algorithms.

I could not see any literature result for this specific situation, but
surely this is something that has been studied.  Some analogous cases for
squares are known to be finite.

Sean.