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Harshad Number

A Positive Integer which is Divisible by the sum of its Digits, also called a Niven Number (Kennedy et al. 1980). The first few are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, ... (Sloane's A005349). Grundman (1994) proved that there is no sequence of more than 20 consecutive Harshad numbers, and found the smallest sequence of 20 consecutive Harshad numbers, each member of which has 44,363,342,786 digits.

Grundman (1994) defined an $n$-Harshad (or $n$-Niven) number to be a Positive Integer which is Divisible by the sum of its digits in base $n\geq 2$. Cai (1996) showed that for $n=2$ or 3, there exists an infinite family of sequences of consecutive $n$-Harshad numbers of length $2n$.


Cai, T. ``On 2-Niven Numbers and 3-Niven Numbers.'' Fib. Quart. 34, 118-120, 1996.

Cooper, C. N. and Kennedy, R. E. ``Chebyshev's Inequality and Natural Density.'' Amer. Math. Monthly 96, 118-124, 1989.

Cooper, C. N. and Kennedy, R. ``On Consecutive Niven Numbers.'' Fib. Quart. 21, 146-151, 1993.

Grundman, H. G. ``Sequences of Consecutive $n$-Niven Numbers.'' Fib. Quart. 32, 174-175, 1994.

Kennedy, R.; Goodman, R.; and Best, C. ``Mathematical Discovery and Niven Numbers.'' MATYC J. 14, 21-25, 1980.

Sloane, N. J. A. Sequence A005349/M0481 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Vardi, I. ``Niven Numbers.'' §2.3 in Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 19 and 28-31, 1991.

© 1996-9 Eric W. Weisstein