Patterns /Number Theory
Highly Divisible Numbers
These numbers have at least a hundred distinct divisors — the opposite extreme from primes, which have exactly two.
Divisor counts come from the exponents in a number's prime factorisation, so reaching a hundred requires several small primes raised to several powers at once. Numbers like 720720 = 2⁴ × 3² × 5 × 7 × 11 × 13 are the result.
This is a demanding threshold: the count above shows how few numbers in the range clear it. Most numbers under a million have fewer than thirty divisors.
Highly divisible numbers are why calendars, clocks and old measurement systems favour 12, 60 and 360 — dividing evenly in many ways is genuinely useful, and genuinely uncommon.
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.
Related number theory patterns
Prime Numbers
A prime has no divisors other than 1 and itself. They are the building blocks every other number is assembled from.
Semiprime Numbers
A semiprime is a product of exactly two primes, counted with multiplicity — 15 = 3 × 5, and 49 = 7 × 7 counts too.
Perfect Squares
A perfect square is some whole number times itself: 144 = 12², 10000 = 100². The range ends neatly on 1,000,000 = 1000².
Perfect Cubes
A perfect cube is a whole number raised to the third power: 216 = 6³, 970299 = 99³. There are only about a hundred in the entire range.
Powers of Two
Each power of two is double the last: 1, 2, 4, 8, 16, and on up through 1024, 65536 and 524288. Doubling exhausts the range in only twenty steps.