Partial answer:

Numbers with at least 12 divisors, and don't divide the product of their first half of factors (minus one) is pretty close. For reference, the sequence is:

84, 96, 108, 132, 150, 156, 198, 200, 204, 220, 228, 234, 260, 272, 276, 294, 304, 306, 312, 340, 342, 348, 368, 372, 380, 392, 408, 414, 444, 456, 460

In particular, if you only consider the first 6 divisors, the set returned seems to be a superset of the list in the question. Of particular interest is the counterexample 294, which has 12 divisors and like 84 is divisible by the first 6 but not the first 5.

Partial answer:

Numbers that don't divide the product of their first six factors, nor first half of factors (minus one) is pretty close. For reference, the sequence is:

84, 96, 108, 132, 150, 156, 198, 200, 204, 220, 228, 234, 260, 272, 276, 294, 304, 306, 312, 340, 342, 348, 368, 372, 380, 392, 408, 414, 444, 456, 460

Code:

from functools import reduce
import operator

def factors(n):    
    return sorted(reduce(list.__add__, 
                ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0)))
def prod(a, i):
    return reduce(operator.mul, a[0:i], 1)

thing = [factors(i) for i in range(2, 5000)]
newset = set()
for a in thing:
    if prod(a, max(5, len(a) // 2 - 1)) % a[-1] != 0:
        newset.add(a[-1])

In particular, if you only consider the first 6 divisors, the set returned seems to be a superset of the list in the question. Of particular interest is the counterexample 294, which has 12 divisors and like 84 is divisible by the first 6 but not the first 5.

The code is the above but replacing the max(5, ...) with 5.

Partial answer:

Numbers with at least 12 divisors, and don't divide the product of their first half of factors (minus one) is pretty close. For reference, the sequence is:

84, 96, 108, 132, 150, 156, 198, 200, 204, 220, 228, 234, 260, 272, 276, 294, 304, 306, 312, 340, 342, 348, 368, 372, 380, 392, 408, 414, 444, 456, 460

Partial answer:

Numbers with at least 12 divisors, and don't divide the product of their first half of factors (minus one) is pretty close. For reference, the sequence is:

84, 96, 108, 132, 150, 156, 198, 200, 204, 220, 228, 234, 260, 272, 276, 294, 304, 306, 312, 340, 342, 348, 368, 372, 380, 392, 408, 414, 444, 456, 460

In particular, if you only consider the first 6 divisors, the set returned seems to be a superset of the list in the question. Of particular interest is the counterexample 294, which has 12 divisors and like 84 is divisible by the first 6 but not the first 5.