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.

Matches in range
95,428
Share of all numbers
9.54%
Out of
1,000,001

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.

5,050 Top <0.01% Singular 1,000,000 Top <0.01% Singular 69,696 Top <0.01% Singular 1,000 Top <0.01% Singular 828,828 Top <0.01% Singular 10,101 Top <0.01% Singular 1,010 Top <0.01% Singular 696,696 Top <0.01% Singular

Smallest examples

The first numbers in the range that match this pattern.

1 2 3 4 5 6 7 8

Related digit arithmetic patterns

Happy Numbers

Square each digit, add them up, repeat. If you eventually reach 1 the number is happy; if not, you fall into a cycle. 19 → 82 → 68 → 100 → 1.
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