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

[Sean A. Irvine]

Hi Allan,

I should have said before, for A343705, I have searched as far as 400^3
with no further solution.

It is precisely the fact that some of these related sequences were missing
from the OEIS that led us to start adding these, and yes there needs to be
an index at some point.  Some are very new (added today!)

https://oeis.org/A344035 Numbers that are the sum of five positive cubes in
exactly four ways.
https://oeis.org/A343988 Numbers that are the sum of four positive cubes in
exactly five ways.
https://oeis.org/A294737 Numbers that are the sum of five positive squares
in exactly three ways (known to be finite)
https://oeis.org/A344244 Numbers that are the sum of five fourth powers in
exactly three ways.
I can probably generate more terms for the first two above and do a graph
like you mention so see if they exhibit a similar pattern.

Hardy and Wright contains a number of results around sequences of these
general type and much appears to be known about the case of squares. I
thought I saw somewhere that every number can be written as a sum of at
most 9 cubes, but I cannot find the reference right now.

Sean.





On Fri, 14 May 2021 at 13:19, Allan Wechsler <acwacw at gmail.com> wrote:

> 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.
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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