[csfodywt] filling the gaps between powers of 2

the integer sequence 3*2^n fits nicely between the gaps between powers of 2.

the sequence 5*2^n fits nicely in the largest gaps (on log scale) in the union of the sequences 2^n and 3*2^n.

the sequence 7*2^n fits nicely in the next largest gaps remaining.

the sequence 9*2^n fits nicely in the next largest gaps remaining.

and so on, with consecutive odd numbers as the multipliers.  verified up to multiplier 19999.

hand-wavy proof: each new odd number ODD*2^0 has not been used in any of the previous sequences, and it sits halfway in between two consecutive even numbers already used.  then, this pattern of nicely "halfway in between" then repeats with everything increased by a factor of 2.

this seems practically useful when you want an exponentially growing integer sequence but do not know a priori how dense you want it.  previously, constructing exponential integer sequences without all terms having easy (smooth) factorizations.