Proximal Fifth Powers That Are Closer To A Cube Than To A Square

[Hans Havermann]

More specifically, Ed Pegg's A117594, where numbers which are cubes  
themselves are trivially excluded. When I first noticed the relative  
proximity of 14706104 and 14706146, which are both members of this  
sequence, I wondered if this was coincidental. I've now run across  
another example: 66430098 and 66430152.

Might there be a number-theoretic explanation?