In number theory, an odious number is a positive integer that has an odd number of $1$s in its binary expansion.

The first odious numbers are:

$1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38...$

Let $a$ denote the increasing sequence of odious numbers where $a_0=1$, $a_1=2$, etc.

Whats the summatory function of the sequence $b$ where $b_i=a_i^k$?

Also: this is useful; but it didn't give out the actual function. Please help.