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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
##### Keywords:
Niven number; Harshad number; digital sum; gaps
OEIS
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