Harshad number

A Harshad number, or Niven number is a natural number. Due to their cross- sum, ie the sum of its digits ( in decimal ) is divisible For example, 777 is divisible by.

The first Harshad Numbers are:

( Sequence A005349 in OEIS )

The above mentioned example with the number 777 can be generalized to all 3- digit natural numbers of the same type:

Any natural number of the form, where any number from 1 to 9 can pose, is a Harshad number, can be divided by sum of its digits so.

The proof follows from the following consideration:

However, now the sum of the digits of. Thus, each natural number of the form is the 37 -fold cross their sum, ie, a Harshad number. q.e.d.

The term Harshad number was introduced by the Indian mathematician DR Kaprekar and is from the Sanskrit word harsha ( "Joy" ) is derived, while Niven number goes back to the mathematician Ivan M. Niven, who described these figures at a conference in 1997.

Swell

• HG 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. Katai, 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