Harshad Numbers (Niven Numbers)

Rule, exact count and rarity

A Harshad number, also called a Niven number, is a positive integer divisible by the sum of its decimal digits. For example, the digit sum of 18 is 9 and 18 ÷ 9 is an integer, so 18 qualifies. The test uses the original digit sum, not repeated digit sums or the digital root. Every positive single-digit integer passes because its digit sum equals itself. Zero is excluded: its digit sum is zero and division by zero is undefined. This property depends on the chosen base, so the atlas specifically uses base ten. Many numbers pass this relatively permissive divisibility test; a Harshad match alone therefore need not imply a particularly high overall rarity score.

The exact count, 95,428, comes from evaluating all 1,000,001 integers from 0 through 1,000,000, including both endpoints. It is not a sample or an estimate. Each matching integer is counted once for this category, even if it contains multiple qualifying segments. Its frequency of 9.54279% describes the chance that a uniformly chosen integer passes this one test. It is not the global score percentile of every member, and overlapping pattern counts must not be added as if they represented disjoint sets.

The pattern supplies −log₂(95,428 / 1,000,001) bits before its within-family weight is applied. Other matched patterns, digit sum, distinct-digit count and short-length signals can change the complete score. Consequently, members of this category need not share a ranking. The highest-scoring representative is 3, with a rarity score of 1,677 and Top 0.0001%. This uses its complete score across all features, not just this pattern. Equal scores are displayed in ascending numerical order.

Example numbers

The highest-scoring match

The highest-scoring representative is 3, with a rarity score of 1,677 and Top 0.0001%. This uses its complete score across all features, not just this pattern. Equal scores are displayed in ascending numerical order.