Harshad number

In recreational mathematics, 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 in base $n$ are also known as $n$-harshad (or $n$-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term “Niven number” arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. All integers between zero and $n$ are $n$-harshad numbers.

Definition

Stated mathematically, let $X$ be a positive integer with $m$ digits when written in base $n$, and let 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 $n$:

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

A number which is a harshad number in every 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 (The number 12 is a harshad number in all bases except octal).

Examples

• The number 18 is a harshad number in base 10, because the sum of the digits 1 and 8 is 9 (1 + 8 = 9), and 18 is divisible by 9 (since 18/9 = 2, and 2 is a whole number).
• The Hardy–Ramanujan number (1729) is a harshad number in base 10, since it is divisible by 19, the sum of its digits (1729 = 19 × 91).
• The number 19 is not a harshad number in base 10, because the sum of the digits 1 and 9 is 10 (1 + 9 = 10), and 19 is not divisible by 10 (since 19/10 = 1.9, and 1.9 is not a whole number).
• Harshad numbers in base 10 form the sequence:

  1, 2, 3, 4, 5, 6, 7, 8, 9, 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, ... (sequence A005349 in the OEIS).

Properties

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 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 or equal to the base number. 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. For example: 11 is not harshad in base 10 because the sum of its digits "11" is 1 + 1 = 2, and 11 is not divisible by 2, while in hexadecimal the number 11 may be represented as "B", the sum of whose digits is also B and clearly B is divisible by B, thus it is harshad in base 16.

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.

Smallest $k$ such that ${\displaystyle k\cdot n}$ is a harshad number are

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 9, 3, 2, 3, 6, 1, 6, 1, 1, 5, 9, 1, 2, 6, 1, 3, 9, 1, 12, 6, 4, 3, 2, 1, 3, 3, 3, 1, 10, 1, 12, 3, 1, 5, 9, 1, 8, 1, 2, 3, 18, 1, 2, 2, 2, 9, 9, 1, 12, 6, 1, 3, 3, 2, 3, 3, 3, 1, 18, 1, 7, 3, 2, 2, 4, 2, 9, 1, ... (sequence A144261 in the OEIS).

Smallest $k$ such that ${\displaystyle k\cdot n}$ is not a harshad number are

11, 7, 5, 4, 3, 11, 2, 2, 11, 13, 1, 8, 1, 1, 1, 1, 1, 161, 1, 8, 5, 1, 1, 4, 1, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 83, 1, 1, 1, 4, 1, 4, 1, 1, 11, 1, 1, 2, 1, 5, 1, 1, 1, 537, 1, 1, 1, 1, 1, 83, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 68, 1, 1, 1, 1, 1, 1, 1, 2, ... (sequence A144262 in the OEIS).

Other bases

The harshad numbers in base 12 are:

1, 2, 3, 4, 5, 6, 7, 8, 9, ᘔ, Ɛ, 10, 1ᘔ, 20, 29, 30, 38, 40, 47, 50, 56, 60, 65, 70, 74, 80, 83, 90, 92, ᘔ0, ᘔ1, Ɛ0, 100, 10ᘔ, 110, 115, 119, 120, 122, 128, 130, 134, 137, 146, 150, 153, 155, 164, 172, 173, 182, 191, 1ᘔ0, 1Ɛ0, 1Ɛᘔ, 200, ...

Smallest $k$ such that ${\displaystyle k\cdot n}$ is a base-12 harshad number are (written in base 10):

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 6, 4, 3, 10, 2, 11, 3, 4, 1, 7, 1, 12, 6, 4, 3, 11, 2, 11, 3, 1, 5, 9, 1, 12, 11, 4, 3, 11, 2, 11, 1, 4, 4, 11, 1, 16, 6, 4, 3, 11, 2, 1, 3, 11, 11, 11, 1, 12, 11, 5, 7, 9, 1, 7, 3, 3, 9, 11, 1, ...

SSmallest $k$ such that ${\displaystyle k\cdot n}$ is not a base-12 harshad number are (written in base 10):

13, 7, 5, 4, 3, 3, 2, 2, 2, 2, 13, 16, 1, 1, 1, 1, 1, 1, 1, 1, 1, 157, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 157, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1885, 1, 1, 1, 1, 1, 3, ...

Consecutive harshad numbers

Maximal runs of consecutive harshad numbers

Cooper and Kennedy proved in 1993 that no 21 consecutive integers are all harshad numbers in base 10.^[1][2] They also constructed infinitely many 20-tuples of consecutive integers that are all 10-harshad numbers, the smallest of which exceeds 10^44363342786.

H. G. Grundman (1994) extended the Cooper and Kennedy result to show that there are 2b but not 2b + 1 consecutive b-harshad numbers.^[2][3] This result was strengthened to show that there are infinitely many runs of 2b consecutive b-harshad numbers for b = 2 or 3 by T. Cai (1996)^[2] and for arbitrary b by Brad Wilson in 1997.^[4]

In binary, there are thus infinitely many runs of four consecutive harshad numbers and in ternary infinitely many runs of six.

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. Given one such suitably chosen sequence, we can convert it to a larger one as follows:

• Inserting zeroes into N will not change the sequence of digital sums (just as 21, 201 and 2001 are all 10-harshad numbers).
• If we insert n zeroes after the first digit, α (worth αbⁱ), we increase the value of N by αbⁱ(bⁿ − 1).
• If we can ensure that bⁿ − 1 is divisible by all digit sums in the sequence, then the divisibility by those sums is maintained.
• If our initial sequence is chosen so that the digit sums are coprime to b, we can solve bⁿ = 1 modulo all those sums.
• If that is not so, but the part of each digit sum not coprime to b divides αbⁱ, then divisibility is still maintained.
• (Unproven) The initial sequence is so chosen.

Thus our initial sequence yields an infinite set of solutions.

First runs of exactly $n$ consecutive 10-harshad numbers

The smallest naturals starting runs of exactly $n$ consecutive 10-harshad numbers (i.e., smallest $x$ such that $x$, $x+1$, ..., ${\displaystyle x+n-1}$ are harshad numbers but $x-1$ and $x+n$ are not) are as follows (sequence A060159 in the OEIS):

$n$	1	2	3	4	5
$x$	12	20	110	510	131052
$n$	6	7	8	9	10
$x$	12751220	10000095	2162049150	124324220	1
$n$	11	12	13	14	15
$x$	920067411130599	43494229746440272890	121003242000074550107423034⋅10^20 − 10	420142032871116091607294⋅10^40 − 4	unknown
$n$	16	17	18	19	20
$x$	50757686696033684694106416498959861492⋅10^280 − 9	14107593985876801556467795907102490773681⋅10^280 − 10	unknown	unknown	unknown

By the previous section, no such $x$ exists for ${\displaystyle n>20}$.

Estimating the density of harshad numbers

If we let $N(X)$ denote the number of harshad numbers ${\displaystyle \leq x}$, then for any given $\epsilon >0$,

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

as shown by Jean-Marie De Koninck and Nicolas Doyon;^[5] furthermore, De Koninck, Doyon and Kátai^[6] 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 for a given number base is an integer $t$ such that there exists some harshad number $N$ whose digit sum is $t$, and $t$, written in that base, terminates $N$ written in the same base.

For example, 18 is a Nivenmorphic number for base 10:

 16218 is a harshad number
 16218 has 18 as digit sum
    18 terminates 16218

Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except 11.^[7]

Multiple harshad numbers

Bloem (2005) defines a multiple harshad number as a harshad number that, when divided by the sum of its digits, produces another harshad number.^[8] He states that 6804 is "MHN-3" on the grounds that

$\begin{array}{l} 6804/18=378\\ 378/18=21\\ 21/3=7 \end{array}$

and went on to show that 2016502858579884466176 is MHN-12. The number 10080000000000 = 1008·10¹⁰, which is smaller, is also MHN-12. In general, 1008·10ⁿ is MHN-(n+2).

References

1. Cooper, Curtis; Kennedy, Robert E. (1993), "On consecutive Niven numbers" (PDF), Fibonacci Quarterly, 31 (2): 146–151, ISSN 0015-0517, Zbl 0776.11003
2. Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. p. 382. ISBN 1-4020-2546-7. Zbl 1079.11001.
3. Grundman, H. G. (1994), "Sequences of consecutive n-Niven numbers" (PDF), Fibonacci Quarterly, 32 (2): 174–175, ISSN 0015-0517, Zbl 0796.11002
4. Wilson, Brad (1997), "Construction of 2n consecutive n-Niven numbers" (PDF), Fibonacci Quarterly, 35: 122–128, ISSN 0015-0517
5. De Koninck, Jean-Marie; Doyon, Nicolas (November 2003), "On the number of Niven numbers up to x", Fibonacci Quarterly, 41 (5): 431–440.
6. De Koninck, Jean-Marie; Doyon, Nicolas; Katái, I. (2003), "On the counting function for the Niven numbers", Acta Arithmetica, 106: 265–275, doi:10.4064/aa106-3-5.
7. Boscaro, Sandro (1996–1997), "Nivenmorphic integers", Journal of Recreational Mathematics, 28 (3): 201–205.
8. Bloem, E. (2005), "Harshad numbers", Journal of Recreational Mathematics, 34 (2): 128.