Harshad Number

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 India. The word "Harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. 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 let the digits be a_i (i = 0, 1, ..., m − 1). (It follows that a_i must be either zero or a positive integer up to n − 1.) X can be expressed as

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

The first 50 Harshad numbers with more than one digit 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, 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.

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.