Harshad number

In mathematics, a Harshad number or Niven number is an integer divisible by the sum of its digits in a given base. These numbers were defined by D. R. Kaprekar, an Indian mathematician. The word "Harshad" It comes from Sanskrit, meaning great joy. Niven's Number takes its name from Ivan Morton Niven, a Canadian and American mathematician, who presented a paper in 1997. All numbers between zero and the base are Harshad numbers.
The first Harshad numbers with more than two digits in base 10 are ((sequence A005349 in OEIS)):

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, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 171, 180, 190, 192, 195, 201 200, 201

Notation

Let X be a positive integer with m digits in base n, and the digits a_i (i = 0, 1,..., m − 1) (It is clear that a_i must be zero or a positive integer up to n) X can be expressed as:

X=␡ ␡ i=0m− − 1aini.{displaystyle X=sum _{i=0}^{m-1}a_{i}n^{i}. !

If there exists an integer A such that the following expression is true, then X is a Harshad number in base n:

X=A␡ ␡ i=0m− − 1ai.{displaystyle X=Asum _{i=0}^{m-1}a_{i}. !

A number that is Harshad in any number base is said to be a total Harshad number or total Niven. There are only four numbers that meet this condition: 1, 2, 4, and 6.

Which 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. For the purpose of determining "how of Hardshard" is a number n, the digits of n can be added only once and n must be divisible by that sum; otherwise, it is not a Harshad number. For example, although 99 is divisible by 9, it turns out that 9 + 9 = 18 and 1 + 8 = 9, which is not a Harshad number, since 9 + 9 = 18 and 99 is not divisible by 18.
The base of the 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 smaller than the base (a one-digit number) or the base number itself. Otherwise, the digits of the prime number will be added to a number that is greater than one but less than the prime number, and obviously will not be divisible.
Although the sequence of factorials begins with base 10 Harshad numbers, they are not all Harshad numbers. 432! is the first number that is not.

Consecutive Harshad numbers

H.G. Grundman proved in 1994 that, in base 10, there are not 21 consecutive integers that are all Harshad numbers. He also found the smallest sequence of 20 consecutive Harshad numbers, which are greater than 10 ^44363342786 .
In binary notation, there are infinitely many sequences of four consecutive Harshad numbers; In ternary, there are infinitely many sequences of six consecutive Harshad numbers. Both facts were proven by T. Cai in 1996.
In general, such maximal sequences range 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. Inserting zeros into N does not change the sum of the digits, so it is possible to convert any solution to a larger one, such as 21, 201, and 2001. Thus, any solution implies an infinite class of solutions.

Estimating the density of Harshad numbers

If N(x) denotes the number of Harshad numbers ≤ x, then for ε > 0,

x1− − ε ε .. N(x).. xlog log xlog x{displaystyle x^{1-varepsilon }ll N(x)ll {frac {xlog log x}{log x}}}}}

as demonstrated by Jean-Marie De Koninck and Nicolas Doyon; Furthermore, De Koninck, Doyon and Kátai showed that,

N(x)=(c+or(1))xlog x{displaystyle N(x)=left(c+o(1)right){frac {x}{log x}}}}}

where c = (14/27) log 10 ≈ 1.1939.