A Harshad number is 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). 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.
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 
.
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.
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21-25, 1980.Sloane, N. J. A. Sequence