Powers of Two

Rule, exact count and rarity

A power of two has the form 2ᵏ for a nonnegative integer exponent k. The sequence begins with 1 because 2⁰ = 1; zero is never a power of two. In binary notation these values have exactly one set bit, followed by zero or more zeros. The engine uses that bit structure to recognize them within its small integer range. Each step doubles the preceding value, so only a limited number fit below a fixed upper bound. Even exponents produce squares, exponents divisible by three produce cubes, and other sequence memberships may overlap. Those relationships explain why the final score varies across the sequence instead of following only the exponent or decimal length.

The exact count, 20, 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 0.002% 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(20 / 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 1, with a rarity score of 1,504 and Top 0.0002%. 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 1, with a rarity score of 1,504 and Top 0.0002%. This uses its complete score across all features, not just this pattern. Equal scores are displayed in ascending numerical order.