# Sequences of consecutive $n$-Niven numbers

• Published in 1992
In the collection
A Niven number is a positive integer that is divisible by the sum of its digits. In 1982, Kennedy showed that there do not exist sequences of more than 21 consecutive Niven numbers. In 1992, Cooper & Kennedy improved this result by proving that there does not exist a sequence of more than 20 consecutive Niven numbers. They also proved that this bound is the best possible by producing an infinite family of sequences of 20 consecutive Niven numbers. For any positive integer $n \gt 2$, define an $n$-Niven number to be a positive integer that is divisible by the sum of the digits in its base $n$ expansion. This paper examines the maximal possible lengths of sequences of consecutive $n$-Niven numbers. The main result is given in the following theorem.

### BibTeX entry

@article{SequencesOfConsecutiveNNivenNumbers,
title = {Sequences of consecutive $n$-Niven numbers},
abstract = {A Niven number is a positive integer that is divisible by the sum of its digits. In 1982, Kennedy showed that there do not exist sequences of more than 21 consecutive Niven numbers. In 1992, Cooper {\&} Kennedy improved this result by proving that there does not exist a sequence of more than 20 consecutive Niven numbers. They also proved that this bound is the best possible by producing an infinite family of sequences of 20 consecutive Niven numbers. For any positive integer $n \gt 2$, define an $n$-Niven number to be a positive integer that is divisible by the sum of the digits in its base $n$ expansion. This paper examines the maximal possible
lengths of sequences of consecutive $n$-Niven numbers. The main result is given in the following theorem. },
url = {http://www.fq.math.ca/Scanned/32-2/grundman.pdf},
author = {H.G. Grundman},
comment = {},
urldate = {2016-12-05},
year = 1992,
collections = {Integerology}
}