Tags

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, …).

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

Pairs where each number equals the sum of the other's proper divisors (220 & 284, 1184 & 1210, 2620 & 2924, …).

Numbers whose divisors have an integer average — d(n) divides σ(n) (1, 3, 5, 6, 7, 11, 13, 14, 15, …).

Numbers whose decimal digits strictly increase, each larger than the one before (123, 1359, 13578).

Numbers whose square ends in the number itself (1, 5, 6, 25, 76, 376, 625, 9376, 90625, …).

Primes exactly midway between their prime neighbors (5, 53, 157, 173, 211, 257, 263, 373, …).

The number of ways to partition a set — 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ….

Numbers whose binary representation reads the same forwards and backwards (5 = 101, 7 = 111, 9 = 1001, 21 = 10101).

The most pieces a cake can be cut into with k planar cuts: (k³+5k+6)/6 → 1, 2, 4, 8, 15, 26, 42, 64, ….

Zusammengesetzte Zahlen, die den kleinen Satz von Fermat für jede zu ihnen teilerfremde Basis erfüllen (561, 1105, 1729, 2465, 2821, 6601, 8911, …).

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

Centered figurate numbers — a point inside nested cubic shells: k³ + (k+1)³ → 1, 9, 35, 91, 189, 341, ….

Centered figurate numbers — a dot ringed by hexagons: 3k(k−1)+1 → 1, 7, 19, 37, 61, 91, ….

Centered figurate numbers — a dot ringed by squares: k² + (k−1)² → 1, 5, 13, 25, 41, 61, 85, ….

Centered figurate numbers — a dot ringed by triangles: (3k²−3k+2)/2 → 1, 4, 10, 19, 31, 46, ….

Primes p where p + 2 is prime or a semiprime (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, …).

Primes that stay prime under every cyclic rotation of their digits (197 → 971 → 719; 1193, 3779, 11939, …).

Numbers where each adjacent digit differs from the next by exactly 1 (1234, 4321, 12321).

Primes of the form 3k²−3k+1 — primes that are also centered hexagonal numbers (7, 19, 37, 61, 127, 271, …).

Numbers not divisible by any perfect cube greater than 1 — every prime appears at most squared (1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, …).

Numbers of the form k·2^k + 1 (3, 9, 25, 65, 161, 385, 897, 2049, …).

Figurate numbers k(4k−3) (1, 10, 27, 52, 85, 126, 175, 232, …).

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

Numbers whose decimal digits strictly decrease, each smaller than the one before (321, 9630, 8531).

Numbers equal to the sum of their digits raised to their position: 89 = 8¹ + 9², 135 = 1¹ + 3² + 5³.

Every digit has a 180° rotational counterpart (0, 1, 8 self-rotate; 6 and 9 swap). A strict superset of strobogrammatic — flippable numbers may rotate to a different number, like 16 ↔ 91.

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

Perfect cubes whose digit sum equals their cube root — there are only six (1, 512, 4913, 5832, 17576, 19683).

Primes whose digit reversal is a different prime (13, 17, 31, 37, 71, 73, 79, 97, 107, 113, …).

Numbers with an even count of 1s in their binary representation (0, 3, 5, 6, 9, 10, 12, 15, …).

Numbers equal to the sum of the factorials of their digits — in base 10 there are only four: 1, 2, 145, 40585.

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

Numbers of the form 2^(2^k) + 1 (3, 5, 17, 257, 65537, 4294967297, …).

Zahlen der Fibonacci-Folge, in der jeder Term die Summe der beiden vorhergehenden ist (0, 1, 1, 2, 3, 5, 8, …).

Numbers whose prime factorization (with exponents) is written with fewer digits than the number itself (125 = 5³, 1024 = 2¹⁰).

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

Numbers of 3+ digits divisible by the two-digit number formed by their first and last digit (100, 105, 108, 110, 120, …).

Iteriertes Summieren der Quadrate der Ziffern führt schließlich zu 1 (1, 7, 10, 13, 19, 23, 28, 31, …).

Numbers considered auspicious in at least one major world culture — Chinese 8 and 9, Western 7, Jewish 18 (chai), etc.

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

Numbers whose sum of divisors exceeds that of every smaller number (1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, …).

Numbers with more divisors than any smaller number — Ramanujan's "anti-primes" (1, 2, 4, 6, 12, 24, 36, 48, 60, 120, …).

A Fibonacci-like sequence with the rule J(n) = J(n−1) + 2·J(n−2): 0, 1, 1, 3, 5, 11, 21, 43, 85, ….

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

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

Digits seed a Fibonacci-like sequence that eventually returns the number itself (14, 19, 28, 47, 61, 75, 197, 742, …).

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

The most pieces a pancake can be cut into with k straight cuts: k(k+1)/2 + 1 → 1, 2, 4, 7, 11, 16, 22, ….

Primes that stay prime as you delete digits from the left (9137 → 137 → 37 → 7). There are exactly 4260.

Numbers of the form x^y + y^x with x ≥ y ≥ 2 (8, 17, 32, 54, 57, 100, 145, 177, 320, 368, …).

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

The common row, column, and diagonal sum of a normal magic square of order k ≥ 3: k(k²+1)/2 → 15, 34, 65, 111, ….

Numbers appearing in a solution of x²+y²+z²=3xyz (1, 2, 5, 13, 29, 34, 89, 169, 194, 233, …).

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

Harshad numbers whose digit-sum quotient is prime: n / digitsum(n) is prime (18, 21, 27, 42, 45, 63, …).

A combinatorial sequence (1, 1, 2, 4, 9, 21, 51, 127, …) counting non-crossing chords, lattice paths, and more.

Numbers equal to the sum of their digits each raised to itself — only 1 and 3435 in base 10.

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

Figurate numbers k(7k−5)/2 (1, 9, 24, 46, 75, 111, 154, 204, …).

Figurate numbers counting points in an octahedron: k(2k²+1)/3 → 1, 6, 19, 44, 85, 146, 231, ….

Numbers with an odd count of 1s in their binary representation (1, 2, 4, 7, 8, 11, 13, 14, …).

The Padovan sequence P(n) = P(n−2) + P(n−3) from seeds 1, 1, 1 (2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, …).

Numbers whose decimal digits read the same forwards and backwards.

Primes that are also decimal palindromes (2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, …).

Numbers containing every decimal digit 0–9 at least once (smallest: 1,023,456,789).

The number of ways to write an integer as a sum of positive integers: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, ….

The Pell sequence: each term is twice the previous plus the one before (1, 2, 5, 12, 29, 70, 169, 408, …).

Figurate numbers stacking pentagons into a pyramid: k²(k+1)/2 → 1, 6, 18, 40, 75, 126, 196, ….

The 4-dimensional simplex figurate numbers: k(k+1)(k+2)(k+3)/24 → 1, 5, 15, 35, 70, 126, 210, ….

Numbers whose count of binary 1s is prime (3, 5, 6, 7, 9, 10, 12, 17, …).

The Perrin sequence P(n) = P(n−2) + P(n−3) from seeds 3, 0, 2 (2, 3, 5, 7, 10, 12, 17, 22, 29, 39, …).

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

Every smaller positive integer is a sum of distinct divisors of n (1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, …).

Standard electronic-component values from the E24 series — an E24 base (10, 11, 12, 13, 15, …) times a power of ten (470, 2.2k, 47k, …).

Primzahlen — natürliche Zahlen größer als 1, deren einzige positive Teiler 1 und sie selbst sind.

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

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

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

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

Primes of the form 4k + 1 — each is a sum of two squares (5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, …).

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

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

Members of Recamán's sequence — a chaotic walk where each step subtracts n if it lands on a positive value not yet visited, otherwise adds n.

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

Numbers divisible by their own number of divisors (1, 2, 8, 9, 12, 18, 24, 36, 40, 56, …). Also called tau numbers.

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 that stay prime as you delete digits from the right (3797 → 379 → 37 → 3). There are exactly 83.

Primes p where (p − 1)/2 is also prime (5, 7, 11, 23, 47, 59, 83, 107, 167, 179, …).

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

Numbers that cannot be written as m + digitsum(m) for any m (1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, …).

Numbers where the digit in each position counts how many times that position's index appears (1210, 2020, 21200, …).

Some subset of the proper divisors sums to the number itself (6, 12, 18, 20, 24, 28, 30, 36, 40, …).

Products of exactly two primes, counted with multiplicity (4, 6, 9, 10, 14, 15, 21, 22, 25, 26, …).

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

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

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

Primes p where 2p + 1 is also prime (2, 3, 5, 11, 23, 29, 41, 53, 83, 89, …).

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

Centered figurate numbers shaped like a six-pointed star: 6k(k−1)+1 → 1, 13, 37, 73, 121, ….

Numbers whose digits form an arithmetic progression with a constant gap of 2 or more (2468, 13579, 147, 9630).

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, …).

Numbers with σ(σ(n)) = 2n (2, 4, 16, 64, 4096, 65536, …) — a second-order cousin of the perfect numbers.

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, …).

Like Fibonacci, but each term sums the previous four: 1, 2, 4, 8, 15, 29, 56, 108, 208, ….

Like Fibonacci but summing the previous three terms (1, 2, 4, 7, 13, 24, 44, 81, 149, 274, …).

Numbers whose digits alternate between two values, ababab… (101, 121, 4747, 69696).

Numbers considered inauspicious in at least one major world culture — Chinese/Japanese 4, Western 13, Italian 17, etc.

Numbers that factor into two "fangs" whose digits are a permutation of the original (1260 = 21·60, 1395 = 15·93).

Zahlen, die gleich der Summe ihrer echten Teiler sind. Nur acht sind unter 2^63 bekannt (6, 28, 496, 8128, …).

Primes of the form (2^p + 1)/3 (3, 11, 43, 683, 2731, 43691, 174763, 2796203, …).

Abundant numbers where no subset of divisors sums to the number (70, 836, 4030, 5830, 7192, 7912, 9272, …).

Primes p with p² dividing 2^(p−1) − 1 — only two are known: 1093 and 3511.

Primes p with p² dividing (p−1)! + 1 — only three are known: 5, 13, and 563.

Numbers of the form k·2^k − 1 (1, 7, 23, 63, 159, 383, 895, 2047, …).

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

Numbers divisible by the product of their digits (1–9, 11, 12, 15, 24, 36, 111, 112, 115, 128, …).

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