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Numbers whose proper divisors sum to more than the number itself.

Powerful numbers that are NOT perfect powers (72, 108, 200, 288, 392, 432, 500, 648, 675, 800, …).

Composite numbers that pass Fermat's little theorem for every base coprime to them (561, 1105, 1729, 2465, 2821, 6601, 8911, …).

The Catalan number sequence (1, 1, 2, 5, 14, 42, 132, …) — appears widely in combinatorics.

Primes p such that p ± 4 is also prime (3, 7, 13, 19, 37, 43, 67, …).

Numbers with hand-written editorial context covering cultural, historical, or technical significance.

Numbers whose proper divisors sum to less than the number itself — the most common kind.

Numbers of the form n! = 1 × 2 × … × n (1, 2, 6, 24, 120, 720, …).

Numbers in the Fibonacci sequence, where each term is the sum of the two preceding ones (0, 1, 1, 2, 3, 5, 8, …).

Iterating the sum of squares of digits eventually reaches 1 (1, 7, 10, 13, 19, 23, 28, 31, 32, 44, …).

Numbers divisible by the sum of their decimal digits (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, …).

Figurate numbers k(5k−3)/2 (1, 7, 18, 34, 55, 81, 112, 148, …).

Figurate numbers k(2k−1) (1, 6, 15, 28, 45, 66, 91, 120, 153, …). Every hexagonal number is also triangular.

Numbers whose square can be split into two parts that sum back to the original (297² = 88209 → 88 + 209 = 297).

Numbers in the Lucas sequence, a Fibonacci-like sequence starting with 2 and 1.

Primes of the form 2^p − 1 where p is also prime (3, 7, 31, 127, 8191, …).

Numbers equal to the sum of their digits each raised to the power of the digit count (153 = 1³ + 5³ + 3³).

Figurate numbers k(3k−2) (1, 8, 21, 40, 65, 96, 133, 176, …).

Numbers whose decimal digits read the same forwards and backwards.

Figurate numbers k(3k−1)/2 (1, 5, 12, 22, 35, 51, 70, 92, …).

Numbers that are the cube of an integer (0, 1, 8, 27, 64, 125, …).

Numbers equal to the sum of their proper divisors. Only eight are known under 2^63 (6, 28, 496, 8128, …).

Numbers that are the square of an integer (0, 1, 4, 9, 16, 25, …).

Numbers of the form 10^k. Basis for decimal place value and metric prefixes.

Numbers of the form 2^k. Fundamental to computing — bytes, bits, address spaces.

Every prime in the factorization appears with exponent ≥ 2 (1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, …).

Prime numbers — natural numbers greater than 1 with no positive divisors other than 1 and themselves.

Numbers of the form k(k+1) — twice a triangular number (0, 2, 6, 12, 20, 30, 42, 56, …).

Numbers where every digit is identical (11, 22, 33, …, 111, 222, … — repunits are a subset).

Numbers consisting only of 1s in decimal: 1, 11, 111, 1,111, 11,111, …

Primes p such that p ± 6 is also prime — “sexy” from the Latin sex (six).

Composite numbers whose digit sum equals the sum of digit sums of their prime factors (4, 22, 27, 58, 85, 94, 121, 166, …).

Products of three distinct primes (30, 42, 66, 70, 78, 102, …).

Figurate numbers k(k+1)(2k+1)/6 — points stacked in a square pyramid (1, 5, 14, 30, 55, 91, 140, 204, 285, …).

Numbers not divisible by the square of any prime — every prime appears in the factorization at most once.

Numbers that look the same when rotated 180° (0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, …).

Figurate numbers k(k+1)(k+2)/6 — the count of balls stacked in a triangular pyramid (1, 4, 10, 20, 35, 56, 84, 120, 165, 220, …).

Numbers of the form k(k+1)/2 — the count of dots forming an equilateral triangle (1, 3, 6, 10, 15, …).

Primes p such that p ± 2 is also prime (3, 5, 7, 11, 13, 17, 19, 29, 31, …).

Numbers in a plausible calendar-year range (1 through 2100).