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Large and small gaps between consecutive Niven numbers. (English) Zbl 1041.11007
A positive integer is said to be a Niven number if it is divisible by the sum of its decimal digits. The authors investigate the occurrence of large and small gaps between consecutive Niven numbers. In fact, let \(n_\ell\) be the smallest positive integer such that the interval \([n, n+\ell-1]\) does not contain any Niven numbers, and let \(T(x)\) denote the number of Niven numbers \(n\leq x\) such that \(n+1\) is also a Niven number. The authors prove that if \(\ell\) is sufficiently large, then \(n_\ell<(100(\ell+2))^{\ell+3}\), and \(T(x)\ll{x\log\log x\over (\log x)^2}\) as \(x\to\infty\).

MSC:
11A63 Radix representation; digital problems
11A25 Arithmetic functions; related numbers; inversion formulas
Software:
OEIS
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