PORC function

Definition
A function $$f$$ on an infinite subset $$S$$ of the natural numbers is termed a Polynomial On Residue Classes function or PORC function if there exists a natural number $$m$$ and polynomials $$f_0, f_1, \dots, f_{m-1}$$ such that if $$n \equiv a \pmod m$$ with $$n \in S$$ and $$0 \le a \le m - 1$$, then $$f(n) = f_a(n)$$.

In other words, the function $$f$$ behaves like a polynomial on each of the residue classes modulo $$m$$.

PORC functions are also called quasipolynomials, though that term has many other meanings in other contexts.

Facts

 * Higman's PORC conjecture: For a fixed natural number $$n$$, define $$f(p,n)$$ for a prime $$p$$ as the number of isomorphism groups of order $$p^n$$. Higman conjectured that for any fixed $$n$$, $$f(p,n)$$ is a PORC function of $$p$$.