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

[Allan Wechsler]

This is fascinating. The first question that came to mind was how far past
2715466 the search had progressed; perhaps Dave Consiglio can give a brief
report.

Next I was moved to look at the graph: it absolutely looks like the supply
of triplex pentacubes is running out. That upsweep sure looks like an
asymptote. That's not dispositive, but it's certainly suggestive.

Perhaps we should look next at the sequence where a(n) is the number of
ways n can be written as the sum of five positive cubes. This is hard to
calculate by hand, and I was surprised not to find a comment in A343705
reading something like "Numbers n for which Axxxxxx(n) = 3." Nor is there a
link to an index of sequences having to do with the number of ways to write
n as a sum of k p'th powers. If that index doesn't exist, it would be
useful.

At any rate, I'm sure that "number of ways to write n as the sum of five
positive cubes" is already in OEIS -- I just can't find it easily. I am
guessing that it shows an inexorable upward trend. The set of
non-pentacubes (A057906), the set of simplex pentacubes (A048926), and the
set of duplex pentacubes (A048927) are all conjectured to be finite. It
might be the case that for all k, the set of k-plex pentacubes is finite.

On Wed, May 12, 2021 at 11:22 PM Sean A. Irvine <sairvin at gmail.com> wrote:

> 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.
>
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