a(n) = Smallest Multiple Of a(n-1) With Exactly n Binary 1's

[Leroy Quet]

Let a(1) = 1. Let a(n) = the smallest positive multiple of a(n-1) with exactly n 1's in its binary representation.

The sequence begins (though, I could be wrong, of course): 1,3,21,105,...

I don't think this sequence is in the EIS, but maybe it is. (Without the 1, a few sequences show up. With the 1, only sequence A034268 matches. But A034268(5) has too many 1's in its binary representation.)

Is the sequence a(n+1)/a(n) (3,7,5,...) in the EIS?

And here is a question that is probably easily answered, but I don't see how to answer it right now:
Can it be proved that there always is a positive multiple of each a(n-1) that has exactly n binary 1's?
Or is the {a(k)} sequence finite?

Thanks,
Leroy Quet