2015 AIME II Problem 6

Steve says to Jon, “I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form P(x)=2 x^{3}-2 a x^{2}+\left(a^{2}-81\right) x-c for some positive integers a and c. Can you tell me the values of a and c ?”

After some calculations, Jon says, “There is more than one such polynomial.”

Steve says, "You’re right. Here is the value of a . " He writes down a positive integer and asks, “Can you tell me the value of c ?”

Jon says, "There are still two possible values of c. "

Find the sum of the two possible values of c.