Run lengths in the binary expansion of n-1 gives the vector of exponents in prime factorization of a(n), with the least significant run corresponding to the exponent of the least prime, 2.

To make this a permutation of nonnegative integers, we subtract one from each run count except for the most significant run, e.g. a(12) = 10, as 12-1 = 11 = 1011 and 10 = 5^1 * 3^(1-1) * 2^(2-1).