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Encyclopedia > Harshad number

A Harshad number, or Niven number, is an integer that is divisible by the sum of its digits in a given number base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word 'Harshad' comes from the Sanskrit language and means '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 numbers between zero and the base number are Harshad numbers. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... A numeral is a symbol or group of symbols that represents a number. ... Shri Dattathreya Ramachandra Kaprekar (January 17, 1905- 1988) was an Indian mathematician, whose name is associated with a number of concepts in number theory. ... The Sanskrit language ( संस्कृता वाक्) is one of the earliest attested members of the Indo-European language family and is not only a classical language, but also an official language of India. ... Ivan Morton Niven (October 25, 1915 – May 9, 1999) was a member of the University of Oregon faculty since 1947 until his retirement in 1981. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... 1997 is a common year starting on Wednesday of the Gregorian calendar. ...

The first few Harshad numbers with more than one digit in base 10 are (sequence A005349 in OEIS): Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

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, 201, 204

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. 10 (ten) is the natural number following 9 and preceding 11. ... 12 (twelve) is the natural number following 11 and preceding 13. ... 18 (eighteen) is the natural number following 17 and preceding 19. ... 20 (twenty) is the natural number following 19 and preceding 21. ... 21 (twenty-one) is the natural number following 20 and preceding 22. ... 24 (twenty-four) is the natural number following 23 and preceding 25. ... 27 (twenty-seven) is the natural number following 26 and preceding 28. ... 30 (thirty) is the natural number following 29 and preceding 31. ... 36 is the natural number following 35 and preceding 37. ... 40 is the natural number following 39 and preceding 41. ... 42 is the natural number following 41 and followed by 43. ... 45 is the natural number following 44 and followed by 46. ... 48 is the natural number following 47 and preceding 49. ... 50 (fifty) is the number following 49 and preceding 51. ... Cardinal fifty-four Ordinal 54th (fifty-fourth) Factorization Divisors 2, 3, 6, 9, 18, 27 Roman numeral LIV Binary 110110 Hexadecimal 36 Fifty-four (54) is the natural number following 53 and preceding 55. ... 60 is the natural number following 59 and preceding 61. ... Sixty-three is a natural number following 62 and preceding 64. ... 70 (seventy) is the natural number following 69 and preceding 71. ... 72 is the natural number following 71 and preceding 73. ... Cardinal eighty Ordinal 80th (eightieth) Numeral system octagesimal Factorization Divisors 2, 4, 5, 8, 10, 16, 20, 40 Roman numeral LXXX Binary 01010000 Hexadecimal 50 80 is the natural number following 79 and preceding 81. ... 81 is the natural number following 80 and preceding 82. ... 84 (eighty-four) is the natural number following 83 and preceding 85. ... 90 is the natural number preceded by 89 and followed by 91. ... 100 (the Roman numeral is C for centum) is the natural number following 99 and preceding 101. ... 102 (one hundred [and] two) is the natural number following 101 and preceding 103. ... 108 is the natural number following 107 and preceding 109. ... 110 (one hundred [and] ten) is the natural number following 109 and preceding 111. ... 111 is the natural number following 110 and preceding 112. ... 112 is the natural number following 111 and preceding 113. ... 114 (one hundred [and] fourteen) is the natural number following 113 and preceding 115. ... 117 is the natural number following 116 and preceding 118. ... 120 (one hundred twenty in American English; one hundred and twenty in British English) is the natural number following 119 and preceding 121. ... 132 is the natural number following 131 and preceding 133. ... 140 is the natural number following 139 and preceding 141. ... 144 is the natural number following 143 and preceding 145. ... 150 is the natural number following 149 and preceding 151. ... 152 (one hundred and fifty-two) is the natural number following 151 and preceding 153. ... One hundred fifty-three is the natural number following one hundred fifty-two and preceding one hundred fifty-four. ... 180 is the natural number following one hundred seventy-nine and preceding one hundred eighty-one. ... 190 is the natural number following one hundred eighty-nine and preceding one hundred ninety-one. ... 200 is the natural number following 199 and preceding 201. ... 1 (one) is the natural number following 0 and preceding 2. ... 2 (two) is the natural number following 1 and preceding 3. ... 4 (four) is the natural number following 3 and preceding 5. ... 6 (six) is the natural number following 5 and preceding 7. ...

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## What numbers can be Harshad numbers? GA_googleFillSlot("encyclopedia_square");

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 = 2 × 32 and 99 is not divisible by 2. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... 99 is the natural number following 98 and preceding 100. ...

Obviously, the base number 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. In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself. ...

In base 10, all factorials are Harshad numbers. In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. ...

## Consecutive Harshad numbers

H.G. Grundman proved in 1994 that in base 10 no 21 consecutive integers are all Harshad numbers. He also found the smallest sequence of 20 consecutive integers that are all Harshad numbers; they exceed 1044363342786. 1994 was a common year starting on Saturday of the Gregorian calendar, and was designated the International year of the Family. ...

In binary there are infinitely many sequences of four consecutive Harshad numbers, while in ternary there are infinitely many sequences of six consecutive Harshad numbers; both of these facts were proven by T. Cai in 1996. In unary base, or tallying, all numbers are Harshad numbers. The binary or base-two numeral system is a system for representing numbers in which a radix of two is used; that is, each digit in a binary numeral may have either of two different values. ... Ternary is the base 3 numeral system. ... 1996 is a leap year starting on Monday of the Gregorian calendar, and was designated the International Year for the Eradication of Poverty. ... Tally marks are a variation of the unary numeral system. ...

## Estimating the density of Harshad numbers

If we let N(x) denote the number of Harshad numbers <= x, then for any given ε > 0,

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

where c = 14/27 log 10 ≈ 1.1939.

## References

• H. G. Grundmann, Sequences of consecutive Niven numbers, Fibonacci Quart. 32 (1994), 174-175
• Jean-Marie De Koninck and Nicolas Doyon, On the number of Niven numbers up to x, Fibonacci Quart. 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

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