IMO 1982 LL POL38

Numbers un,k (1 \leqk \leqn) are defined as follows:

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IMO 1982 LL POL38

Origin: POL

Problem

Numbers un,k (1 \leqk \leqn) are defined as follows: u1,1 = 1, un,k = n k  −  d|n, d|k, d>1 un/d,k/d (the empty sum is defined to be equal to zero). Prove that n | un,k for every natural number n and for every k (1 \leqk \leqn).