Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes/Historical Note
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Historical Note on Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes
This result can be found in Curious and Interesting Numbers by David Wells from $1986$, but it is a mystery as to why it was presented.
There appears to be absolutely nothing special about $351 \, 120$, as there are many cubes far smaller which possess the same property.
It is possible that Wells misinterpreted something that caught his eye, but googling for properties of $351 \, 120$ reveals that it is spectacularly uninteresting.