Numbers that can be written as a sum of three cubes in exactly one way (a^3 + b^3 + c^3) 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? 
 A: 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.
A: 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
A: 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

