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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a harshad number (or Niven number) in a given number base is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
that is divisible by the sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

from
India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...

. The word "harshad" comes from the
Sanskrit Sanskrit (; attributively , ; nominalization, nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had Trans-cul ...

' (joy) + ' (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

in 1977.

# Definition

Stated mathematically, let be a positive integer with digits when written in base , and let the digits be $a_i$ ($i = 0, 1, \ldots, m-1$). (It follows that $a_i$ must be either zero or a positive integer up to .) can be expressed as :$X=\sum_^ a_i n^i.$ is a harshad number in base if: :$X \equiv 0 \bmod .$ 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: , , , and . The number is a harshad number in all bases except
octal The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, uses a base-10 numbe ...

.

# 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 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

by 9. * 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. *In base 10, every natural number expressible in the form 9Rnan, where the number Rn consists of n copies of the single digit 1, n>0, and an is a positive integer less than 10n and multiple of n, is a harshad number. (R. D’Amico, 2019). The number 9R3a3 = 521478, where R3 = 111, n = 3 and a3 = 3×174 = 522, is a harshad number; in fact, we have: 521478/(5+2+1+4+7+8) = 521478/27 = 19314. *Harshad numbers in
base 10 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

form the sequence: *: , , , , , , , , , , , 18, 20, 21, 24, 27, 30, , 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112,
114 114 may refer to: *114 (number) 114 (one hundred ndfourteen) is the natural number following 113 (number), 113 and preceding 115 (number), 115. In mathematics *114 is an abundant number, a sphenic number and a Harshad number. It is the sum of th ...
, 117, 120, 126,
132 132 may refer to: *132 (number) 132 (one hundred ndthirty-two) is the natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''t ...
, 133, 135, 140, 144, 150, 152, 153,
156 Year 156 ( CLVI) was a leap year starting on Wednesday A leap year starting on Wednesday is any year with 366 days (i.e. it includes 29 February February 29, also known as leap day or leap year day, is a date added to leap years. A le ...
, 162,
171 Year 171 ( CLXXI) was a common year starting on Monday A common year starting on Monday is any non-leap year (i.e., a year with 365 days) that begins on Monday, 1 January, and ends on Monday, 31 December. Its dominical letter hence is G. The ...
, 180,
190 Year 190 (Roman numerals, CXC) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Sura (or, less frequently, year 943 ''Ab ...
, 192, 195, 198, 200, ... . *All integers between
zero 0 (zero) is a number, and the numerical digit used to represent that number in numeral system, numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. A ...

# Properties

Given the divisibility test for , 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 , the digits of can only be added up once and 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. All numbers whose base ''b'' digit sum divides ''b''−1 are harshad numbers in base ''b''. For a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
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 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
base 12 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using 12 (number), twelve as its radix, base. The number twelve (that is, the number written as "12" in the decimal numerical syste ...
the number 11 may be represented as “Ɛ”, the sum of whose digits is also Ɛ. Since Ɛ is divisible by itself, it is harshad in base 12. Although the sequence of
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
s starts with harshad numbers in base 10, not all factorials are harshad numbers. 432! is the first that is not. (432! has digit sum = 3897 = 32×433 in base 10, thus not dividing 432!) Smallest such that $k \cdot n$ 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, ... . Smallest such that $k \cdot n$ 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, ... .

# Other bases

base 12 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using 12 (number), twelve as its radix, base. The number twelve (that is, the number written as "12" in the decimal numerical syste ...
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, ... where ᘔ represents ten and Ɛ represents eleven. Smallest such that $k \cdot n$ 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, ... Smallest such that $k \cdot n$ 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, ... Similar to base 10, not all factorials are harshad numbers in base 12. After 7! (= 5040 = 2Ɛ00 in base 12, with digit sum 13 in base 12, and 13 does not divide 7!), 1276! is the next that is not. (1276! has digit sum = 14201 = 11×1291 in base 12, thus does not divide 1276!)

## Maximal runs of consecutive harshad numbers

Cooper and Kennedy proved in 1993 that no 21 consecutive integers are all harshad numbers in base 10. They also constructed infinitely many 20-tuples of consecutive integers that are all 10-harshad numbers, the smallest of which exceeds 1044363342786. extended the Cooper and Kennedy result to show that there are 2''b'' but not 2''b'' + 1 consecutive ''b''-harshad numbers. This result was strengthened to show that there are infinitely many runs of 2''b'' consecutive ''b''-harshad numbers for ''b'' = 2 or 3 by and for arbitrary ''b'' by Brad Wilson in 1997. In
binary Binary may refer to: Science and technology Mathematics * Binary number A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typical ...
, 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''·''bk'' − ''b'' to ''N''·''bk'' + (''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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
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 consecutive 10-harshad numbers

The smallest naturals starting runs of ''exactly'' consecutive 10-harshad numbers (i.e., smallest such that $x, x+1, \cdots, x+n-1$ are harshad numbers but $x-1$ and $x+n$ are not) are as follows :
style="text-align:right;" , - , , , 1 , , 2 , , 3 , , 4 , , 5 , - , , , 12 , , 20 , , 110 , , 510 , , , - , , , 6 , , 7 , , 8 , , 9 , , 10 , - , , , , , , , , , , , 1 , - , , , 11 , , 12 , , 13 , , 14 , , 15 , - , , , , , , , , , , , unknown , - , , , 16 , , 17 , , 18 , , 19 , , 20 , - , , , , , , , unknown , , unknown , , unknown , -
By the previous section, no such exists for $n > 20$.

# Estimating the density of harshad numbers

If we let $N\left(x\right)$ denote the number of harshad numbers $\le x$, then for any given $\epsilon > 0$, :$x^ \ll N\left(x\right) \ll \frac$ as shown by Jean-Marie De Koninck and Nicolas Doyon; furthermore, De Koninck, Doyon and Kátai proved that :$N\left(x\right)=\left(c+o\left(1\right)\right)\frac,$ where $c = \left(14/27\right) \log 10 \approx 1.1939$ and the $o\left(1\right)$ term uses
Big O notation Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
.

Every natural number not exceeding one billion is either a harshad number or the sum of two harshad numbers. Conditional to a technical hypothesis on the zeros of certain Dedekind zeta functions, Sanna proved that there exists a positive integer $k$ such that every natural number is the sum of at most $k$ harshad numbers, that is, the set of harshad numbers is an additive basis. The number of ways that each natural number 1, 2, 3, ... can be written as sum of two harshad numbers is: :0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 3, 2, 4, 3, 3, 4, 3, 3, 5, 3, 4, 5, 4, 4, 7, 4, 5, 6, 5, 3, 7, 4, 4, 6, 4, 2, 7, 3, 4, 5, 4, 3, 7, 3, 4, 5, 4, 3, 8, 3, 4, 6, 3, 3, 6, 2, 5, 6, 5, 3, 8, 4, 4, 6, ... . The smallest number that can be written in exactly 1, 2, 3, ... ways as the sum of two harshad numbers is: :2, 4, 6, 8, 10, 51, 48, 72, 108, 126, 90, 138, 144, 120, 198, 162, 210, 216, 315, 240, 234, 306, 252, 372, 270, 546, 360, 342, 444, 414, 468, 420, 642, 450, 522, 540, 924, 612, 600, 666, 630, 888, 930, 756, 840, 882, 936, 972, 1098, 1215, 1026, 1212, 1080, ... .

# Nivenmorphic numbers

A Nivenmorphic number or harshadmorphic number for a given number base is an integer such that there exists some harshad number whose digit sum is , and , written in that base, terminates 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. In fact, for an even integer ''n'' > 1, all positive integers except ''n''+1 are Nivenmorphic numbers for base ''n'', and for an odd integer ''n'' > 1, all positive integers are Nivenmorphic numbers for base ''n''. e.g. the Nivenmorphic numbers in
base 12 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using 12 (number), twelve as its radix, base. The number twelve (that is, the number written as "12" in the decimal numerical syste ...
are (all positive integers except 13). The smallest number with base 10 digit sum ''n'' and terminates ''n'' written in base 10 are: (0 if no such number exists) :1, 2, 3, 4, 5, 6, 7, 8, 9, 910, 0, 912, 11713, 6314, 915, 3616, 15317, 918, 17119, 9920, 18921, 9922, 82823, 19824, 9925, 46826, 18927, 18928, 78329, 99930, 585931, 388832, 1098933, 198934, 289835, 99936, 99937, 478838, 198939, 1999840, 2988941, 2979942, 2979943, 999944, 999945, 4698946, 4779947, 2998848, 2998849, 9999950, ...

defines a multiple harshad number as a harshad number that, when divided by the sum of its digits, produces another harshad number.. He states that 6804 is "MHN-4" on the grounds that :$\begin 6804/18&=378\\ 378/18&=21\\ 21/3&=7\\ 7/7&=1 \end$ (it is not MHN-5 since $1/1=1$, but 1 is not "another" harshad number) and went on to show that 2016502858579884466176 is MHN-12. The number 10080000000000 = 1008·1010, which is smaller, is also MHN-12. In general, 1008·10''n'' is MHN-(''n''+2).