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f(mn^2) = f(mn) +a f(n), if prime p|n, then f(p) \ne 0 and f(p)|f(n)

Source: 2008 Indonesia TST stage 2 test 4 p4

December 15, 2020
number theoryfunctional equationfunctionalfunctional equation in N

Problem Statement

Find all pairs of positive integer α\alpha and function f:N→N0f : N \to N_0 that satisfies (i) f(mn2)=f(mn)+αf(n)f(mn^2) = f(mn) + \alpha f(n) for all positive integers m,nm, n. (ii) If nn is a positive integer and pp is a prime number with p∣np|n, then f(p)≠0f(p) \ne 0 and f(p)∣f(n)f(p)|f(n).
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