— Burt Kaliski Jr. (@modulomathy) September 9, 2013
There are two perfect numbers that are small enough to divide a day-number, 6 and 28, and they both divide 252.
Because the least common multiple of 6 and 28 is 84, any day-number divisible by 84 will be divisible by both of these perfect numbers. Because 6 and 28 are the only perfect numbers small enough to divide a day-number (the next one is 496), the answer to the question consists of just the day-numbers other than 252 that are divisible by 84, which are 84, 168, and 336.
Recall that a perfect number is a number the sum of whose smaller divisors add up to itself:
6 = 1 + 2 + 3
28 = 1 + 2 + 3 + 7 + 14
It is not known whether there are infinitely many perfect numbers, but all the ones that are known are even and have the form (2p-1)*2p-1 where both p and 2p-1 are prime (i.e., 2p-1 is a Mersenne prime). (To the reader: Which values of p produce 6 and 28?)