#math trivia for #February13: 44 is the fifth number of the form p^2*q for distinct primes p, q. What are p and q? What were the first four?

— Burt Kaliski Jr. (@modulomathy) February 13, 2012

Factoring the day’s number, 44, quickly yields the values of *p* and *q*:

44 = 2^2 * 11 ,

giving *p* = 2, *q* = 11.

Because *p* and *q* are distinct, it follows that three of the first four numbers of this form involve *p* = 2 and the three odd primes less than 11:

12 = 2^2 * 3

20 = 2^2 * 5

28 = 2^2 * 7 .

The remaining number must involve p = 3 and q = 2:

18 = 3^2 * 2 .

As a final check that 44 is indeed the fifth such number, consider that the next number with *p* = 3 is 45, and the first with *p* = 5 is 50 — so we’ve covered all the possibilities less than 44.

Bonus question: How many other numbers of the form *p*^{2}*q *will occur this year (i.e., up to 366)?