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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 Indiamarker. 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 ai (i = 0, 1, ..., m − 1). (It follows that ai 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 1044363342786.

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 · bk - b to N · bk + (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



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