The first few elements of the sequence are given by
$1, 2, 3, 4, 5, 7, 8, 9, 11, \ldots$
and we note that the first two positive integers not contained in the sequence are $6$ and $10$.
These would not have made the lcm larger when it was "their turn", and they would certainly
not do so at a later date. We therefore note that a number that has been "left out" in the
sequence of positive integers cannot turn up at some later point.

Next, we note that each prime is included in the sequence. When a prime number $p$ is the next number under consideration, its inclusion will certainly always increase the value of the lcm, and it will therefore certainly be included in the sequence $\langle a_n \rangle$. This is also true of any power of a prime. All powers of primes are therefore included in the sequence $\langle a_n \rangle$.

On the other hand, any integer $N$ that has at least two different prime divisors has only divisors of the form $p^k$ (with $p$ prime) that have already been included in the sequence of powers of primes to that point. Inclusion of $N$ in the sequence will therefore certainly not raise the value of the lcm, and no such $N$ can be included in the sequence $\langle a_n \rangle$.

Summarizing, we note that the sequence $\langle a_n \rangle$ is composed of the number $1$ and all powers of primes in ascending order.
qed