Try to set maximum border of the answer for every big N (for example, you can exclude all primes > N/2 etc.)
After that think whether this maximum value can be achieved (as I remember, it is possible only if numbers with less count of divisors should apeear first in resulting sequence - it's not hard to prove).
Such thinking allows you to set some numbers in the beginning of the sequence. Other 40-45 values can be found by backtrack.

seems to be true

Of course.