Patterns /Digit Arithmetic
Harshad Numbers
A Harshad — or Niven — number is divisible by the sum of its own digits. 18 works because 1 + 8 = 9 and 18 ÷ 9 = 2.
How long might harshad numbers take?
1 in 10.5 per roll · 10 rolls on average.
At one daily specimen a day, the mean is about 0 years. A 50% chance takes 7 rolls; 90% takes 23.
7 daily rolls
50.44%
30 daily rolls
95.06%
365 daily rolls
100.00%
95.06% chance of at least one match in 30 independent rolls.
Waiting times are averages, not deadlines. Previous misses do not increase the next roll’s chance. Only Daily Specimens stamp your Atlas.
Harshad numbers link a number's written form to its arithmetic, which is unusual: most divisibility properties ignore notation entirely. This one only makes sense in base ten.
They thin out as numbers grow, because digit sums rise roughly logarithmically while the numbers themselves rise linearly — so the chance of an exact division steadily falls. The count above is the exact answer across the range.
Every repdigit is automatically a Harshad number, which is one reason the strongest numbers on the leaderboard tend to collect traits from several groups at once.
Highest-scoring examples
The best-scoring numbers of four digits or more that carry this pattern. Shorter numbers are excluded here because their length alone dominates any score ranking.
Smallest examples
The first numbers in the range that match this pattern.