[seqfan] A possible characterization of A125121

[Allan Wechsler]

Are these also exactly those integers whose binary representations "tile"
the integers?

Represent an integer N as a set of indices S={i} such that sum 2^i = N.
 For any N we can ask: does there exist a family of shifted copies of S
which are (a) disjoint, and (b) whose union covers the integers?  Each N
poses a pleasant little tiling problem.

I amused myself by listing the integers that tile, and found that it
matched A125121, whose given characterization also involves binary
representations, but seems quite different from my tiling description.

Can anyone prove that A125121 contains exactly the integers that tile, or
find a counterexample?