1.It depends on the  number of factors the Door ‘i’ has(including 1 and i ).for example,Door Number having odd number of Factors ,  then it would be          Opened.else it will be Closed.
Ex 1:76 – factors are-1, 2, 4, 19, 38, 76…even(6)factors it has .. so 76th door will be closed..
Ex 2:64- factors are- 1,2,4,8,16,32,64..odd(7) factors has..so 64th door will be opened
or
2.Only Perfect square doors will be open …

Felice and Nicolas

numbers that follow  odd and sum of digits is a prime are :11,21,23,25,29,41,43,49,61,65,67,83,85.
among these numbers follows “difference of digits also prime” are : 25,29,41,47,49,61,83,85.
last but not least , product of digits is same as one of it’s  digits means one digit should be 1.
So Answer is 41 or 61.