Perfect Cube Numbers

Rule, exact count and rarity

A perfect cube equals k × k × k for a nonnegative integer k. The dataset includes 0 and 1, while negative cubes lie outside the allowed input range. The engine checks the nearest integer cube root by cubing it back, so membership is based on an exact integer equality. Consecutive cubes spread apart faster than consecutive squares, leaving a relatively small set of cube values within the bounded range. Some cubes are also squares: nonnegative sixth powers satisfy both definitions. The corresponding signals share a factorization family and are not simply added as fully independent events. Decimal appearances such as repeated digits are tested separately, so equally valid cubes can have very different total rarity scores.

The exact count, 101, 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 0.0101% 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(101 / 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.