A positive integer which is divisible by the sum of its digits, also called a Niven number (Kennedy
et al. 1980) or a multidigital number (Kaprekar 1955). The first few are 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, ... (OEIS 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
Grundman (1994) defined an -Harshad (or -Niven) number to be a positive
integer which is divisible by the sum of its digits
in base .
Cai (1996) showed that for or 3, there exists an infinite family of sequences of consecutive
-Harshad numbers of length .
Define an all-Harshad (or all-Niven) number as a positive integer which is divisible by the sum of its digits in all bases . Then only 1, 2, 4, and 6 are all-Harshad numbers.
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R. E. "Chebyshev's Inequality and Natural Density." Amer. Math.
Monthly96, 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 -Niven Numbers." Fib. Quart.32, 174-175,
1994.Kaprekar, D. R. "Multidigital Numbers." Scripta
Math.21, 27, 1955.Kennedy, R. E. and Cooper, C. N.
"On the Natural Density of the Niven Numbers." Abstract 816-11-219, Abstracts
Amer. Math. Soc.6, 17, 1985.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 "The On-Line Encyclopedia of Integer Sequences."Vardi,
I. "Niven Numbers." §2.3 in Computational
Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 19
and 28-31, 1991.Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 171, 1986.