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Multiples of repunits as sum of powers of ten. (English) Zbl 1300.11014
Summary: The sequence $$P_{k,n}=1+10^k+10^{2k}+\cdots+10^{(n-1)k}$$ can be used to generate infinitely many Smith numbers with the help of a set of suitable multipliers. We prove the existence of such a set, consisting of constant multiples of repunits, that generalizes to any value of $$k\geqslant 9$$. This fact complements the earlier results which have been established for $$k\leqslant 9$$.