Triangular Numbers

Rule, exact count and rarity

A triangular number is a sum of the first k positive integers, giving k(k + 1)/2. This atlas also permits k = 0, so zero is included as the empty sum. One useful membership test is that 8n + 1 must be an odd perfect square; the engine checks its square root for integer status. Successive triangular numbers differ by the next positive integer, so the gaps grow gradually. Triangular membership describes an arithmetic sequence of values rather than a shape in the written digits. A triangular number can also be a square or satisfy other factorization and decimal patterns. These extra matches help determine the final score, while the triangular frequency is computed across the complete inclusive range.

The exact count, 1,414, 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 0.1414% 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(1,414 / 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.