Mersenne Numbers (Binary All Ones)

Rule, exact count and rarity

A binary all-ones number has the form 2ᵏ − 1, called a Mersenne number, with k at least two for this badge. The binary digits then contain nothing but 1s. The minimum length excludes 1, even though some broader sequence conventions include it. A Mersenne number is not necessarily a Mersenne prime: 15 has binary representation 1111 but is composite. Every qualifying value is odd and is also a binary palindrome. These related representation properties share a scoring family, so their information contributions are discounted for overlap. The values nearly double at successive lengths, leaving only a small set inside a fixed upper bound, while other arithmetic matches determine differences in their overall ranking.

The exact count, 18, 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 0.0018% 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(18 / 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.