Numbers with at Least 100 Divisors

Rule, exact count and rarity

This category requires at least 100 distinct positive divisors. For a positive integer with prime factorization p₁ᵃ¹ × p₂ᵃ² × …, the divisor count is (a₁ + 1)(a₂ + 1)… . The engine computes that count from prime exponents and applies the stated threshold. This is not the standard definition of a highly composite number: that term concerns record divisor counts relative to all smaller positive integers. A threshold match need not set a record. Zero is excluded by the engine's factorization convention, rather than being treated as having an infinite count. A large divisor count can coexist with many other properties, but the badge records only whether the threshold is reached.

The exact count, 2,809, 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 0.2809% 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(2,809 / 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 828,828, with a rarity score of 991 and Top 0.0049%. 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 828,828, with a rarity score of 991 and Top 0.0049%. This uses its complete score across all features, not just this pattern. Equal scores are displayed in ascending numerical order.