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?

$\begingroup$ 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

2$\begingroup$ 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
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.
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.asahinet.or.jp/~KC2HMSM/mathland/math04/cube00.htm
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

1$\begingroup$ According to the paper mentioned at the top of the table (Elsenhans & Jahnel, New sums of three cubes, Math. Comp. 78 (2009), 12271230), 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