I wrote a program that finds the sequence of such integers and was surprised to find that is https://oeis.org/A217856 (Numbers with three prime factors, not necessarily distinct, except cubes of primes.) I imagined that as $n$ grew larger the number of such composite divisors would slowly grow, but this appears not to be the case. I have checked up to n=500000.
For example, 12 is the first number in the sequence because the divisors of 12 are 2,3,4 and 6. Now ignore the prime divisors and examine the composites that remain, 4 and 6. 4 does not divide 6, so 12 is in the sequence.
Why can't a number with more prime divisors have more such composite divisors?