Is it too late for April Fools?

Algebra Level 5

Let N\mathbb{N} be the set of nonnegative integers. Define a function f:N3Nf : \mathbb{N}^3 \to \mathbb{N} satisfying the following for any a,b,cN{0}a,b,c \in \mathbb{N} \setminus \{0\}:

{f(0,0,0)=2015f(a,0,0)=f(a1,1,0)f(0,b,0)=f(0,b1,1)f(0,0,c)=f(c1,0,0)f(a,b,0)=f(a1,b1,2)f(a,0,c)=f(c1,1,a1)f(0,b,c)=f(0,b1,c+1)f(a,b,c)=f(b,c,a1)\begin{cases} f(0,0,0) &= 2015 \\ f(a,0,0) &= f(a-1,1,0) \\ f(0,b,0) &= f(0,b-1,1) \\ f(0,0,c) &= f(c-1,0,0) \\ f(a,b,0) &= f(a-1,b-1,2) \\ f(a,0,c) &= f(c-1,1,a-1) \\ f(0,b,c) &= f(0,b-1,c+1) \\ f(a,b,c) &= f(b,c,a-1) \\ \end{cases}

A happinization of a positive integer nn is an infinite sequence of integers, where the first term of the sequence is nn, and every term afterwards is the sum of the squares of the digits of the previous term. For example, the happinization of 20152015 is 2015,30,9,81,65,61,37,58,89,145,20,4,16,37,2015, 30, 9, 81, 65, 61, 37, 58, 89, 145, 20, 4, 16, 37, \ldots. A happy number is a positive integer whose happinization ends with an infinite number of 11s; all other positive integers are sad numbers. For example, 20152015 above is sad, but 11 (with happinization 1,1,1,1,1, 1, 1, 1, \ldots) is happy.

For any positive integer nn, define pnp_n as the product of the first n!n! happy numbers. Let A=p1,F=p2,I=p3,L=p4,O=p5,P=p6,R=p7,S=p8A = p_1, F = p_2, I = p_3, L = p_4, O = p_5, P = p_6, R = p_7, S = p_8. Compute the value of f(APRIL,FOOLS,2015)f(APRIL,FOOLS,2015).

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