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Part of 2009 Czech-Polish-Slovak Match

Problems(1)

Infinite sequence with exactly 2009 different numbers

Source: Czech-Polish-Slovak Match, 2009

8/21/2011
For positive integers aa and kk, define the sequence a1,a2,…a_1,a_2,\ldots by a1=a,andan+1=an+k⋅ϱ(an)for n=1,2,…a_1=a,\qquad\text{and}\qquad a_{n+1}=a_n+k\cdot\varrho(a_n)\qquad\text{for } n=1,2,\ldots where ϱ(m)\varrho(m) denotes the product of the decimal digits of mm (for example, ϱ(413)=12\varrho(413)=12 and ϱ(308)=0\varrho(308)=0). Prove that there are positive integers aa and kk for which the sequence a1,a2,…a_1,a_2,\ldots contains exactly 20092009 different numbers.
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