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sequence with sum of digits

Source: All Russian MO 2015, grade 10, problem 4

August 8, 2015
number theorySequence

Problem Statement

We denote by S(k)S(k) the sum of digits of a positive integer number kk. We say that the positive integer aa is nn-good, if there is a sequence of positive integers a0a_0, a1,…,ana_1, \dots , a_n, so that an=aa_n = a and ai+1=ai−S(ai)a_{i + 1} = a_i -S (a_i) for all i=0,1,...,n−1i = 0, 1,. . . , n-1. Is it true that for any positive integer nn there exists a positive integer bb, which is nn-good, but not (n+1)(n + 1)-good? A. Antropov
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