Kaprekar’s other discoveries
Harshad numbers and self numbers
4 min read · Updated 19 September 2026
Kaprekar was fascinated by one simple operation: adding up the digits of a number. Two of the families he named come directly from it. Harshad numbers are divisible by their digit sum. Self numbers cannot be produced by adding any number to its own digit sum. Both are easy to test by hand and lead to surprisingly deep questions.
Harshad numbers
A harshad number is divisible by the sum of its digits. 18 is harshad because 1 + 8 = 9 and 18 ÷ 9 = 2. 19 is not, because 1 + 9 = 10 does not divide 19.
Kaprekar chose the name from Sanskrit, where it is usually translated as “joy-giver”. In English-language literature they are also called Niven numbers, after the mathematician Ivan Niven, who discussed them at a conference in 1977.
The harshad numbers below 60 are:
1–10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54
Is 6174 a harshad number?
Yes. Its digits add to 6 + 1 + 7 + 4 = 18, and 6174 ÷ 18 = 343, which is 7³. Kaprekar’s constant is therefore also a harshad number. Its three-digit sibling 495 has the same digit sum, 18, but 495 ÷ 18 = 27.5, so 495 is not.
Every multiple of 9 has a digit sum that is a multiple of 9, and as explained in why every number reaches 6174, every Kaprekar result is a multiple of 9. That makes harshad results more likely along Kaprekar paths than among random numbers.
Runs of harshad numbers
Harshad numbers are common among small numbers but become rarer as numbers grow. One much-studied question is how many can appear in a row. In base ten there can be at most twenty consecutive harshad numbers, a result published by Cooper and Kennedy in 1993. The first run of that length begins at an enormous number.
Self numbers
Now reverse the question. Define a generator: take any number m and add its digit sum. From 15 you get 15 + 1 + 5 = 21, so 15 generates 21. A self number is one that no m can generate.
The first self numbers are:
1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, …
Notice the steps of 11 from 20 up to 97. They appear because adding a number to its digit sum tends to jump over values in a regular rhythm. Kaprekar studied these numbers from Devlali, and they are sometimes called Devlali numbers in his honour.
How to test a number
To check whether n is a self number, you only need to try generators close to n. A number with k digits has a digit sum of at most 9k, so any generator of n lies between n − 9k and n. For 20, test 11 to 19: 11 gives 13, 12 gives 15, … 19 gives 29. None produces 20, so 20 is a self number.
- Count the digits k of your number n.
- Test every m from n − 9k up to n − 1.
- Compute m + digit sum of m for each.
- If none equals n, then n is a self number.
Harshad in every base
Change the base and a different list of harshad numbers appears. A handful of numbers, however, are harshad in every base: 1, 2, 4 and 6. Checking 6 in base 4, for instance, gives the digits 1 and 2, whose sum 3 divides 6. No other number has this property, a result that shows how quickly base-dependent patterns break down.
Why these ideas matter
Both families depend on base ten: change the base and the lists change completely. They are a good introduction to the idea that some properties belong to numbers themselves, such as being prime, while others belong to the way we write numbers down. Kaprekar’s constant is firmly in the second group, which is one reason why claims that 6174 appears in nature deserve caution.