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Based on online info, it seems that most of these numbers have many solutions. Are there any that have only 1 known solution or only a few solutions?

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  • $\begingroup$ Do you mean "numbers that can be written as a sum of cubed integers in exactly one way"? $\endgroup$ – LSpice Jan 5 at 4:22
  • $\begingroup$ Yes that is what I meant thank you. $\endgroup$ – Aza Jan 5 at 4:46
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    $\begingroup$ Do you mean cubes of positive integers? $\endgroup$ – François Brunault Jan 5 at 9:37
  • $\begingroup$ Related: math.stackexchange.com/q/88776 $\endgroup$ – Watson Jan 5 at 14:20
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Yes, (with the positivity assumption) this is OEIS A025395, and members of this sequence are also found in a table on MathWorld with links to similar sequences, such as those positive integers which may be written as the sum of two cubes in exactly two ways which $1729=1^3+12^3=9^3+10^3$, famously, is an example of.

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Regarding equation $n=a^3+b^3+c^3$.

And required by 'OP' are value's of $(n,a,b,c)$

While the link(OEIS A05395) given by Josiah gives value of 'n'

it does not give the values for $(a,b,c)$. While the 'Mathworld link

gives only a few numerical solutions.

For numerous values of $(n,a,b,c)$ 'OP' can go to

the link below on Seiji Tomita website:

http://www.maroon.dti.ne.jp/fermat/eindex.html

And click on article # 163 in the section for third powers.

Also the below mentioned links could be use full:

http://arxiv.org/pdf/1604.07746

http://cr.yp.to/threecubes.html

http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/cube00.htm

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According to the table prepared by Elsenhans & Jahenel there are some integers $n$ which have only one solution.

Some of the examples in said table for $n<999$ are

453,564,660,663,822,912,966,978

Remarkably all the above examples are divisible by three. So there seems to be a pattern.

Explicitly the solutions (n,a,b,c) are shown below:

n   | a           | b          |  c
----+-------------+------------+------------
453 |          10 |         13 |          -14
564 | 53872419107 |-1300749634 | -53872166335
660 |      228487 |    -159116 |      -199163
663 |     1068938 |    -105841 |     -1068592
822 |          22 |        -17 |           17
912 |    55956937 |  -14232281 |    -55648340
966 |        2548 |       -965 |        -2501
978 |       40169 |       8666 |       -40303
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    $\begingroup$ According to the paper mentioned at the top of the table (Elsenhans & Jahnel, New sums of three cubes, Math. Comp. 78 (2009), 1227-1230), the authors determined all solutions with a, b, c bounded in absolute value by 10^14. So if there is only one solution in the table, this does not mean that there is only one solution. In fact, as far as I know, the expectation is that there should always be infinitely many solutions unless n is congruent to 4 or 5 mod 9, when there are none (because there are none mod 9). $\endgroup$ – Michael Stoll Jan 14 at 11:47

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