Sequence Derived From Coprimality

[Leroy Quet]

(sent to sci.math as well)

Let a[1] = 1;

Let, for n >= 2, a[n] be the minimum positive integer not equal to
(a[1],a[2],...a[n-1]), where GCD(a[n],a[m]) = 1 for all m where
ceiling(n/2) <= m <= n-1.

This sequence begins (I think, figured by hand):

1, 2, 3, 5, 4, 7, 9, 11, 13, 17, 8, 19, 23, 25, ...

This may be easy to prove, actually, but is this sequence
made up only of powers of primes?

Is there a direct means of generating the sequence?

Anything else? (such as, is this sequence a rearrangement of the
prime-powers?)

Thanks,
Leroy Quet