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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 \(k\leqslant 9\).
MSC:
11A63 Radix representation; digital problems
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References:
[1] McDaniel, W. L., The existence of infinitely many k-Smith numbers, Fibonacci Quart., 25, 76-80, (1987) · Zbl 0608.10012
[2] Wilansky, A., Smith numbers, Two-Year College Math. J., 13, 21, (1982)
[3] Witno, A., A family of sequences generating Smith numbers, J. Integer Seq., 16, (2013), Art. 13.4.6 · Zbl 1285.11020
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