A Harshad number is a positive integer which is divisible by the sum of its decimal 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). Cooper and Kennedy (1993) proved that there is no sequence of more than 20 consecutive Harshad numbers and constructed a sequence of 20 consecutive Harshad numbers whose members have 4436342786 digits. Wilson (1996) constructed a smaller example with 1788 digits per member.
A constructive family is given by
 |
for positive integers 
and 
satisfying 
(D'Amico 2019, p. 22). Each such number has
digit sum 
, which divides it. Indeed, for
an integer 
with 
,
 |
and the two blocks of 
decimal digits, including leading zeros, have corresponding
digits summing to 9. Taking 
gives the stated family.
Grundman (1994) defined an 
-Harshad (or 
-Niven) number to be a positive
integer which is divisible by the sum of its digits in base 
, where 
is an integer. Cai (1996)
showed that for 
or 3, there exists an infinite family of sequences of consecutive 
-Harshad numbers of length 
.
An all-Harshad (or all-Niven) number is 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.
-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.Numberphile. "2016502858579884466176." Sep. 24,
2025.