#math trivia for #September9: #252 is divisible by two perfect numbers. What other day-numbers (1-365) have this property?

— 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 (2* ^{p}*-1)*2

^{p-1}where both

*p*and 2

*-1 are prime (i.e., 2*

^{p}*-1 is a Mersenne prime). (To the reader: Which values of*

^{p}*p*produce 6 and 28?)