[3] Harshad numbers were defined by D. R. Kaprekar, a mathematician from India.
[4] The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver.
The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977.
X can be expressed as X is a harshad number in base n if:
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: 11 is not harshad in base 10 because the sum of its digits “11” is 1 + 1 = 2, and 11 is not divisible by 2; while in base 12 the number 11 may be represented as “B”, the sum of whose digits is also B.
Although the sequence of factorials starts with harshad numbers in base 10, not all factorials are harshad numbers.
has digit sum 3897 = 32 × 433 in base 10, thus not dividing 432!)
is not a harshad number are The harshad numbers in base 12 are: where A represents ten and B represents eleven.
is a base-12 harshad number are (written in base 10): Smallest k such that
has digit sum 14201 = 11 × 1291 in base 12, thus does not divide 1276!)
Cooper and Kennedy proved in 1993 that no 21 consecutive integers are all harshad numbers in base 10.
[6][7] They also constructed infinitely many 20-tuples of consecutive integers that are all 10-harshad numbers, the smallest of which exceeds 1044363342786.
H. G. Grundman (1994) extended the Cooper and Kennedy result to show that there are 2b but not 2b + 1 consecutive b-harshad numbers for any base b.
[7][8] This result was strengthened to show that there are infinitely many runs of 2b consecutive b-harshad numbers for b = 2 or 3 by T. Cai (1996)[7] and for arbitrary b by Brad Wilson in 1997.
[9] In binary, there are thus infinitely many runs of four consecutive harshad numbers and in ternary infinitely many runs of six.
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.
Given one such suitably chosen sequence, we can convert it to a larger one as follows: Thus our initial sequence yields an infinite set of solutions.
are not) are as follows (sequence A060159 in the OEIS): By the previous section, no such x exists for
as shown by Jean-Marie De Koninck and Nicolas Doyon;[10] furthermore, De Koninck, Doyon and Kátai[11] proved that where
Conditional to a technical hypothesis on the zeros of certain Dedekind zeta functions, Sanna proved that there exists a positive integer
For example, 18 is a Nivenmorphic number for base 10: Sandro Boscaro determined that for base 10 all positive integers are Nivenmorphic numbers except 11.
Weisstein, Eric W. "Harshad Number".