A

**Harshad number**, or

**Niven number**
in a given

number base, is an

integer that is divisible by the

sum of its digits when written in that base.
Harshad numbers were defined by

D.
R. Kaprekar, a
mathematician from India. The
word "Harshad" comes from the

Sanskrit
, meaning "great
joy". The Niven numbers take their name from

Ivan M. Niven
from a paper delivered at a conference on

number theory in 1997. All integers between

zero and

*n* are Harshad numbers
in base

*n*.

Stated mathematically, let

*X* be a positive integer with

*m* digits when written in base

*n*, and the digits
be

*a*_{i} (

*i* = 0, 1, ...,

*m* − 1). (It follows that

*a*_{i}
must be either zero or a positive integer up to

*n* − 1.)

*X* can be expressed as

- X=\sum_{i=0}^{m-1} a_i n^i.

If there exists an integer

*A* such that the following
holds, then

*X* is a Harshad number in base

*n*:

- X=A\sum_{i=0}^{m-1} a_i.

The first 50 Harshad numbers with more than one digit in

base 10 are :

- 10, 12,
18, 20,
21, 24,
27, 30,
36, 40,
42, 45,
48, 50,
54, 60,
63, 70,
72, 80,
81, 84,
90, 100,
102, 108,
110, 111,
112, 114,
117, 120,
126, 132,
133, 135,
140, 144,
150, 152,
153, 156,
162, 171,
180, 190,
192, 195,
198, 200.

A number which is a Harshad number in any number base is called an

**all-Harshad number**, or an

**all-Niven
number**. There are only four all-Harshad numbers:

1,

2,

4, and

6.

## What numbers can be Harshad numbers?

Given the

divisibility test for

9, one might be tempted to generalize
that all numbers divisible by 9 are also Harshad numbers. But for
the purpose of determining the Harshadness of

*n*, the
digits of

*n* can only be added up once and

*n* must
be divisible by that sum; otherwise, it is not a Harshad number.
For example,

99, although divisible by 9
as shown by 9 + 9 = 18 and 1 + 8 = 9, is not a Harshad number,
since 9 + 9 = 18, and 99 is not divisible by 18.

The base number (and furthermore, its powers) will always be a
Harshad number in its own base, since it will be represented as
"10" and 1 + 0 = 1.

For a

prime number to also be a Harshad
number, it must be less than the base number, (that is, a 1-digit
number) or the base number itself. Otherwise, the digits of the
prime will add up to a number that is more than 1 but less than the
prime, and obviously, it will not be divisible.

Although the sequence of

factorials starts
with Harshad numbers in base 10, not all factorials are Harshad
numbers. 432! is the first that is not.

## Consecutive Harshad numbers

H.G. Grundman proved in 1994 that, in base 10,
no 21 consecutive integers are all Harshad numbers. She also found
the smallest 20 consecutive integers that are all Harshad numbers;
they exceed 10

^{44363342786}.

In

binary, there are
infinitely many sequences of four consecutive Harshad numbers; in

ternary, there are infinitely
many sequences of six consecutive Harshad numbers. Both of these
facts were proven by T. Cai in 1996.

In general, such maximal sequences run from

*N · b*^{k} -
b to

*N · b*^{k} + (

*b*-1), where

*b* is the base,

*k* is a relatively large power, and
N is a constant. Interpolating zeroes into N will not change the
sequence of digital sums, so it is possible to convert any solution
into a larger one by interpolating a suitable number of zeroes,
just as 21 and 201 and 2001 are all Harshad numbers base 10. Thus
any solution implies an infinite class of solutions.

## Estimating the density of Harshad numbers

If we let

*N*(

*x*) denote the number of Harshad
numbers ≤ x, then for any given ε > 0,

- x^{1-\varepsilon} \ll N(x) \ll \frac{x\log\log x}{\log x}

as shown by

Jean-Marie De
Koninck and Nicolas Doyon; furthermore, De Koninck, Doyon and
Kátai proved that

- N(x)=(c+o(1))\frac{x}{\log x}

where

*c* = (14/27) log 10 ≈ 1.1939.

## Nivenmorphic numbers

A

**Nivenmorphic number** or

**Harshadmorphic
number** is an integer

*t* such that written in a
given number base, it is possible to find a Harshad number

*Nt*, whose

digit sum is

*t*, and

*t* terminates

*Nt* written in the
same number base.

For example, 18 is a Nivenmorphic number:

16218 has 18 as digital sum
18 divides 16218
18 terminates 16218

Sandro Boscaro determined that all base-10 integers are
Nivenmorphic numbers except

11.

## References

- H. G. Grundmann,
*Sequences of consecutive Niven
numbers*, Fibonacci
Quarterly 32 (1994), 174-175
- Jean-Marie De Koninck and Nicolas Doyon,
*On the number of
Niven numbers up to x*, Fibonacci Quarterly Volume 41.5
(November 2003), 431–440
- Jean-Marie De Koninck, Nicolas Doyon and I. Katái,
*On the
counting function for the Niven numbers*, Acta Arithmetica 106 (2003), 265–275
- Sandro Boscaro,
*Nivenmorphic Integers*, Journal of Recreational
Mathematics **28**, 3 (1996 - 1997): 201–205