nonHarshad

A nonHarshad number in base 10 is a positive integer that is not divisible by the sum of its decimal digits. The sum of digits function, denoted s(n), is the sum of the digits of n in base 10. Numbers that are divisible by s(n) are called Harshad numbers (also known as Niven numbers); nonHarshad numbers are therefore the complement of Harshad numbers among positive integers.

Examples illustrate the distinction. The number 11 has digit sum s(11) = 2, and 11 mod 2 = 1, so 11 is nonHarshad. Similarly, 14 has s(14) = 5 and 14 mod 5 = 4, making it nonHarshad. Other small nonHarshad examples include 22 (s = 4, 22 mod 4 = 2) and 23 (s = 5, 23 mod 5 = 3). By contrast, numbers like 18 (s = 9; 18 mod 9 = 0) and 12 (s = 3; 12 mod 3 = 0) are Harshad.

A notable property is that there are infinitely many nonHarshad numbers. In particular, every prime number greater than 9 is nonHarshad. If p is a prime with p > 9, then s(p) is at least 2 and at most 9d (where d is the number of digits of p), and s(p) cannot divide p because p is prime and s(p) is neither 1 nor p. Consequently, primes beyond 9 cannot be Harshad, reinforcing that nonHarshad numbers are abundant.

In practice, nonHarshad is used to describe numbers not divisible by the sum of their digits in base 10, distinguishing them from Harshad/Niven numbers. The concept is base-specific; a number may be Harshad in another base even if it is nonHarshad in base 10.