In

_{n}a_{n}, where the number R_{n} consists of n copies of the single digit 1, n>0, and a_{n} is a positive integer less than 10^{n} and multiple of n, is a harshad number. (R. D’Amico, 2019). The number 9R_{3}a_{3} = 521478, where R_{3} = 111, n = 3 and a_{3} = 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 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,

^{2}×433 in base 10, thus not dividing 432!)
Smallest such that $k\; \backslash cdot\; n$ 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 $k\; \backslash cdot\; n$ 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, ... .

^{44363342786}.
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, 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 ''αb^{i}''), we increase the value of ''N'' by ''αb^{i}''(''b^{n}'' − 1).
* If we can ensure that ''b^{n}'' − 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 ''b^{n}'' = 1 modulo all those sums.
* If that is not so, but the part of each digit sum not coprime to ''b'' divides ''αb^{i}'', then divisibility is still maintained.
* ''(Unproven)'' The initial sequence is so chosen.
Thus our initial sequence yields an infinite set of solutions.

^{10}, which is smaller, is also MHN-12. In general, 1008·10^{''n''} is MHN-(''n''+2).

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 languag ...

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
Dattatreya Ramchandra Kaprekar ( mr, दत्तात्रेय रामचंद्र कापरेकर; 17 January 1905 – 1986) was an Indian recreational mathematician who described several classes of natural numbers incl ...

, 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, structure, space, models, and change.
History
On ...

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 , ; 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-cultural diffusion ...

' (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 integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

in 1977.
Definition

Stated mathematically, let be a positive integer with digits when written in base , and let the digits be $a\_i$ ($i\; =\; 0,\; 1,\; \backslash ldots,\; m-1$). (It follows that $a\_i$ must be either zero or a positive integer up to .) can be expressed as :$X=\backslash sum\_^\; a\_i\; n^i.$ is a harshad number in base if: :$X\; \backslash equiv\; 0\; \backslash bmod\; .$ 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: 1, 2, 4, and 6. The number 12 is a harshad number in all bases exceptoctal
The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7. This is to say that 10octal represents eight and 100octal represents sixty-four. However, English, like most languages, ...

.
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. * 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 9R102 102 may refer to:
*102 (number), the number
* AD 102, a year in the 2nd century AD
*102 BC, a year in the 2nd century BC
* 102 (ambulance service), an emergency medical transport service in Uttar Pradesh, India
* 102 (Clyde) Field Squadron, Royal En ...

, 108, 110, 111, 112, 114 114 may refer to:
*114 (number)
*AD 114
*114 BC
*114 (1st London) Army Engineer Regiment, Royal Engineers, an English military unit
*114 (Antrim Artillery) Field Squadron, Royal Engineers, a Northern Irish military unit
*114 (MBTA bus)
*114 (New Je ...

, 117 117 may refer to:
*117 (number)
*AD 117
*117 BC
*117 (emergency telephone number)
*117 (MBTA bus)
* 117 (TFL bus)
*117 (New Jersey bus)
*''117°'', a 1998 album by Izzy Stradlin
*No. 117 (SPARTAN-II soldier ID), personal name John, the Master Chief ...

, 120, 126, 132 132 may refer to:
*132 (number)
*AD 132
*132 BC
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Year 132 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Laenas and Rupilius (or, less frequently, year 622 ''Ab urbe condita'') ...

, 133 133 may refer to:
*133 (number)
*AD 133
*133 BC
*133 (song) 133 may refer to:
*133 (number)
*AD 133
*133 BC
__NOTOC__
Year 133 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Scaevola ...

, 135 135 may refer to:
*135 (number)
*AD 135
*135 BC
*135 film, better known as 35 mm film, is a format of photographic film used for still photography
*135 (New Jersey bus) 135 may refer to:
*135 (number)
*AD 135
*135 BC
*135 film
135 film, mor ...

, 140, 144 144 may refer to:
* 144 (number), the natural number following 143 and preceding 145
* AD 144, a year of the Julian calendar, in the second century AD
* 144 BC, a year of the pre-Julian Roman calendar
* ''144'' (film), a 2015 Indian comedy
* ''14 ...

, 150 150 may refer to:
*150 (number), a natural number
*AD 150, a year in the 2nd century AD
*150 BC, a year in the 2nd century BC
*150 Regiment RLC
*Combined Task Force 150
See also
* List of highways numbered 150
The following highways are numbered ...

, 152, 153, 156, 162, 171
Year 171 ( CLXXI) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Herennianus (or, less frequently, year 924 '' Ab urbe c ...

, 180
__NOTOC__
Year 180 ( CLXXX) was a leap year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Rusticus and Condianus (or, less frequently, year 933 ''Ab ...

, 190, 192, 195, 198, 200, ... .
*All integers between zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...

and are -harshad numbers.
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 , 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 aprime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...

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 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 factorials 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 = 3Other 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, ... where ᘔ represents ten and Ɛ represents eleven. Smallest such that $k\; \backslash cdot\; n$ 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 $k\; \backslash cdot\; n$ 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!)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. They also constructed infinitely many 20-tuples of consecutive integers that are all 10-harshad numbers, the smallest of which exceeds 10First runs of exactly consecutive 10-harshad numbers

The smallest naturals starting runs of ''exactly'' consecutive 10-harshad numbers (i.e., smallest such that $x,\; x+1,\; \backslash cdots,\; x+n-1$ 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 $N(x)$ denote the number of harshad numbers $\backslash le\; x$, then for any given $\backslash epsilon\; >\; 0$, :$x^\; \backslash ll\; N(x)\; \backslash ll\; \backslash frac$ as shown by Jean-Marie De Koninck and Nicolas Doyon; furthermore, De Koninck, Doyon and Kátai proved that :$N(x)=(c+o(1))\backslash frac,$ where $c\; =\; (14/27)\; \backslash log\; 10\; \backslash approx\; 1.1939$ and the $o(1)$ term usesBig O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund L ...

.
Sums of harshad numbers

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 anadditive basis In additive number theory, an additive basis is a set S of natural numbers with the property that, for some finite number k, every natural number can be expressed as a sum of k or fewer elements of S. That is, the sumset of k copies of S consists o ...

.
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 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, ...Multiple harshad numbers

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 :$\backslash begin\; 6804/18\&=378\backslash \backslash \; 378/18\&=21\backslash \backslash \; 21/3\&=7\backslash \backslash \; 7/7\&=1\; \backslash end$ (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 = 1008·10References

External links

{{Divisor classes Base-dependent integer sequences