Peter's arithmetic

Peter has recently been fascinated by numbers that can be expressed in the form \( f(a, b, c) = a(b-c)^3+b(c-a)^3+c(a-b)^3\) for some integers \(a,b\) and \(c\). He thinks that we should use this system to do arithmetic in future.

How many positive integers \( N \leq 1000 \) can be expressed in the form of \( f(a, b, c) \)?

Details and assumptions

As a specific example, since \( f(-1, 2, 3) = 48\), hence \(48\) can be expressed in the form of \( f(a, b, c) \).

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