MathDB

A

is a non-empty subset of positive integers. Let

f(A)=\{abc-b-c+2\vert a,b,c\in A\}

Determine all integers

n

greater than

1

so that we can divide the set of positive integers into

A_1, A_2, \dots, A_n

(

A_i\neq \emptyset (i=1, 2, \dots , n)

\forall 1\le i < j \le n, A_i\cap A_j = \emptyset

and

\bigcup_{i=1}^{n} A_i=\mathbb{N}^*

) satisfy that

\forall 1\le i\le n, f(A_i) \subseteq A_i

Problem Statement