Let P(x) be a function that returns the product of digits of x. For example P(935) = 9*3*5 = 135. Now let us define another function P_repeat(x):

int P_repeat(int x){
    if(x < 10) return x
    return P_repeat(P(x))
}

For a value v, find the number of numbers x less than N such that P_repeat(x) = v.

How would I make a transition between states in a digit DP for this problem?

Please help, and thanks in advance!

UPD: The bounds are N <= 10^13.