This puppy sparked a big debate for us. Take a swing at it

[Troop]

     The answer is 1.

     I say this because of what is stated in the problem.

     "it was difficult for farmer Joe to find the perfect pumpkin"

     The question states that farmer joe was looking for "the" perfect pumpkin so I think that there is only one.

     My brain hurts. 

[Vortex]

None. There is no such thing as a perfect pumpkin.

[Eowyn]

Given that every 6th pumpkin is "an every 3rd pumpkin," you can eliminate any 6th pumpkin calculations (this is where Ali is on the money).  The idea of sieving out multiples of 3, 4, and 5 has already been executed by Hoffs, Righty, Ali, etc.  528 is most definitely the answer.

If you want to picture this without drawing a table of 1320 elements, you can reduce the sample size to 3*4*5, or 60, and then further to 30 by noticing repitition in this reduced set.  Conveniently, and necessarily, 30 is exactly 1/44 of 1320.  Sieving out multiples of 3, 4, and 5 out of 30 numbers is trivial and can be drawn up in no time.  This leaves 12 elements.  Our original set is 1320 pumpkins, and sieving the pumpkins out of 1320 is 12 * 44, or 528.

You made my head hurt.