Almanac of Interesting Numbers


largest negative integer



value assigned to the divergent infinite series 1+2+3+4+ by Ramanujan summation



cardinality of the empty set

the additive identity

was represented using a shell in the ancient Mayan civilisation


real part of all non-trivial zeros of the Riemann zeta function – if the Riemann hypothesis is true!


Euler–Mascheroni constant γ

the limiting difference between the harmonic series and the natural logarithm


Catalan’s constant G

the result of the infinite series 112132+152172+

unknown whether irrational or not


the multiplicative identity

the only natural number that isn’t prime or composite

wasn’t considered a “number” by many ancient and medieval mathematicians



proportion between the frequencies of adjacent semitones in the 12 tone equal temperament scale


Apéry’s constant ζ3

sum of the reciprocals of the positive cubes: 113+123+133+

appears in quantum electrodynamics and random spanning trees



can’t be constructed using ruler a compass, which was one of the three famous geometric problems of antiquity



first known irrational number, discovered by Pythagoras or one of his students

ratio of diagonal and the side length in a square

proportion between the sides of DIN paper sizes


Golden ratio φ

the larger of the two real roots of x2=x+1

ratio of the side and one of the diagonals in a pentagon



the measure of the fish

length of the space diagonal of a cube with edge length 1

height of an equilateral triangle with side length 2


smallest and only even prime number

Euler characteristic of a polyhedron homeomorphic to a sphere



length of the diagonal of a 1 × 2 rectangle


second Feigenbaum constant α

ratio between the width of a tine and the width of one of its two subtines


Euler’s number e

limit of 1+1nn as n approaches infinity

unique positive number a such that the graph of the function y=ax has a slope of 1 at x=0


number of dimensions we can see

number of particles generations in the standard model

first Mersenne prime


Pi or π

circumference of a circle with diameter 1

transcendental number, which means that the famous ancient problem of squaring a circle is impossible

closely approximated by 227 or 355113


smallest composite number

maximum number of colours needed to colour any map in the Euclidean plane

number of Nucleobase types in the DNA: A, G, C, T

unlucky number in Chinese, Japanese and Korean culture


first Feigenbaum constant δ

limiting ratio of consecutive bifurcation intervals in a one-parameter logistic map


number of Platonic solids

number of vertices of a pyramid

number of vertices in the smallest non-planar (K5)

the cycle of fifths underlies to musical harmonies

fifth Fibonacci number


smallest perfect number

order of the smallest non-abelian group

number of distinct Quarks

number of regular polytopes


Tau τ=2π

circumference of a circle with radius 1

period of the sin, cos and tan functions


number of vertices of the smallest polygon that can’t be constructed with ruler and compass

number of hills in Rome and wonders of the ancient world

secret agent number of James Bond

number of days in a week


largest Fibonacci Number which is a cube

order of the smallest non-commutative unitary ring

number of tentacles of an octopus

number of bits in a byte

lucky number in Chinese culture: its name sounds similar to “prosperity”


maximum number of cubes needed to sum to any positive integer

number of muses in Greek mythology

exponential factorial, since 9=321

number of significant points in a triangle that lie on a circle: the Nine-Point circle


base of our number system

house number of the British Prime Minister

number of Jewish and Christian commandments

sum of the first three primes, first four integers and first four factorials


smallest palindrome in base-10

number of space-time dimensions in M-theory

first Apollo mission to land of the moon

number of players in a football team


only sublime number less than 1 trillion

smallest abundant number

number of months per year and signs in the Zodiac

number of faces of the dodecahedron


number of different Archimedean solids

number of cards of a suit

unlucky number in Western culture


sum of the first three squares, i.e. a square-pyramidal number

open meandric number: the number of meanders in non-self-intersecting oriented curves

number of pounds in one stone

fourth Catalan number


magic constant in a 3×3 magic square.

largest composite number with one one group of that order

triangular, hexagonal, pentatope and Bell number


only number of the form xy=yx with x and y distinct integers

number of pawns in a chess set, and each player starts with 16 pieces

length of credit card numbers


number of wallpaper groups

“least random number” or Feller Number

sum of the first 4 prime numbers, and only prime which is the some of 4 consecutive primes


only non-zero number that is twice the sum of its digits.

sum of the first three pentagonal numbers, making it a “pentagonal pyramidal number”

lucky number and the value for life in Jewish numerology


maximum number of 4th powers needed to sum to any number

number of cells in the only non-trivial magic hexagon


number of rooted trees with 6 vertices

number of faces of an icosahedorn and vertices of a dodecahedron

number of quarter or half turns required to optimally solve a Rubik’s cube in the worst case

base of the ancient Mayan number system

sum of the first 4 triangular numbers, and thus a tetrahedral number


magic number in Blackjack

number of dots on a 6-sided die

smallest number of distinct squares into which a square can be dissected

smallest Fibonacci number whose digits and digit sums are also Fibonacci


number of partitions of 8

number of Major Arcana cards in divinatory Tarot

pentagonal and a centred heptagonal number


least number of distinct integer-sided cuboids needed to make up another cuboid

number of unsolved problems posed by Hilbert in 1900

smallest odd non-twin prime

number of people required so that the probability of a shared birthday is greater than 50%


Gelfond’s constant eπ


largest number divisible by all integers less than its square root

only non-trivial integer n with the property that 12++n2 is a perfect square

number of hours in a day


smallest square that can be written as a sum of two squares.

automorphic, aspiring and non-sociable number


number of sporadic groups

only positive number to be directly between a square and a cube

smallest non-palindromic number whose square is a palindrome: 262=676


sum of the digits of its cube: 273=19683

in the Collaz Conjecture you need 112 steps to get from 27 to 1


second perfect number

harmonic divisor number, happy number and triangular number

number of days in the lunar cycle

number of dominoes in a standard double-six set


largest Prime that is the sum of three consecutive square numbers: 4 + 9 + 16 = 29

Saturn requires over 29 years to orbit the Sun


sum of the first four squares

largest number such that all smaller numbers coprime to it are prime

number of edges of the icosahedron and dodecahedron

smallest sphenic number


centered triangular, pentagonal and decagonal number

Mersenne, lucky and supersingular prime

number of musical triads (12 major, 12 minor, 4 diminished, and 3 augmented)

most common number of days per month

11111 in base 2


ninth happy number

smallest nontrivial fifth power


sum of the totient functions of the first ten integers

freezing point of water at sea level in Fahrenheit


largest integer that isn’t the sum of distinct triangular numbers

number of vertebrae in a human spine

sum of the first four factorials


smallest number that has the same number of divisors as the previous and following number

magic number of the order four magic square

nontotient and noncototient number


number of different hexominoes

sum of the first five triangular numbers

highly cototient and tetrahedral number


smallest non-trivial number which is both square and triangular

sum of the cubes of the first three integers.

the sum of the first 36 integers is 666


maximum number of 5th powers needed to sum to any number

lucky, irregular, unique and cuban prime

human body temperature in degrees Celsius

number of plays written by Shakespeare


magic number of an order 3 magic hexagon

number of slots in American Roulette

largest even number which cannot be written as the sum of two odd composite numbers


sum of five consecutive primes: 3 + 5 + 7 + 11 + 13

sum the first three powers of three: 3 + 9 + 27

number of steps in the title of a Hitchcock film


only number whose letters are in alphabetical order

Venus returns to the same point in the night sky every 40 years

octagonal number

number of days in Lent


the polynomial n2+n+41 gives primes for n<40

lucky number of Euler, and the largest such prime

number of Mozart’s last symphony


answer to the Ultimate Question of Life, the Universe, and Everything, according to Deep Thought

magic number in a 3×3×3 magic cube

fifth Catalan number


Lowest atomic number of any element without stable isotopes

smallest prime that is not a Chen prime

smallest prime number expressible as the sum of two, three, four or five primes


number of derangements of 5 items

largest number of regions the plane can be divided into by 7 circles

Tribonacci, happy and octahedral number


third Kaprekar number, since 452=2025 and 20+25=45

triangular, hexagonal and 16-gonal number


number of human Chromosomes

Erdős–Woods number

Wedderburn-Etherington number


largest number of cubes that cannot tile a cube

atomic number of silver

not a palindrome in any base b for 2b45

Carol number


double factorial of 6

number of Ptolemaic tessellations

highly composite number and the smallest number with 10 divisors


multiplying the period of 149 by integers coprime to 49 permutes the digits

type 149 in your calculator and you will be very surprised…


smallest number that can be written as the sum of of two squares in two distinct ways: 50=52+52=72+12

represented by the letter L in Roman numerals


number of ways to draw non-intersecting lines between six points on the boundary of a circle

both a pentagonal number and a centred pentagonal number

number of a highly classified United States Air Force facility


number of playing cards in a standard deck (4 suits with 13 cards each)

number of white keys on a piano

untouchable number, because it is never the sum of the proper divisors of any other number


can be written as 35 (reversed) in hexadecimal, a property shared by many multiples of 53

smallest prime number that does not divide the order of any sporadic group


number of coloured squares on a Rubik’s cube

perfect round on a par 72 golf course


largest number that is both triangular and Fibonacci

sum of the first 10 integers

square pyramidal number, since it is the sum of the first 5 square numbers


maximum determinant of an 8*8 matrix of zeroes and ones

sum of the first six triangular numbers or the first six odd primes

tetranacci and pronic number


can be written as 111 in base 7

Leyland number, since 25+52=57

the 57-cell is a self-dual abstract regular four-dimensional polytope.


number of degrees in 1 radian


number of commutative semigroups of order 4

sum of the first seven Prime numbers


number of stellations of an icosahedron

number of orthorhombic space groups


order of the smallest non-abelian simple group

base of the Babylonian number system

number of minutes in one hour and one degree


appears three times in the list of Fortunate numbers

maybe the largest prime dividing the product of the next two primes plus 1


only number whose cube, 238328, consists of three digits each occurring two times

written as 222 in base 5


number of partially ordered sets of 5 elements


number of tiles on a chess board

superperfect number with σσn=2n

maximum number of strokes in any Chinese character

smallest number with 7 divisors

index of Graham’s number in the sequence 3, 27, 7625597484987, …


smallest number that becomes a square when its reverse is either added to or subtracted from it


sum of the first 22 integers

largest number below 2000 not containing an ‘e’ (an eban number)


smallest number that is palindromic in bases 5 and 6

maximal number of letters in website domains

irregular and lucky and Pillai prime


truncated Tetrahedral Number

number of tetragonal space groups

in a normal distribution, 68% of values are within one standard deviation of the mean


its square and its cube together contain every digit exactly once

largest factorial on many calculators


smallest weird number: an abundant number which is not the sum of any subset of its divisors

a Pell number and a generalised heptagonal number


algebraic degree of Conway’s constant

divides the sum of all primes less than it

largest prime factor of the order of any sporadic simple group


maximum number of spheres that can touch each other in a lattice packing in 6 dimensions

average number of heartbeats per minute

smallest Achilles number


least number of sixths powers needed to sum to every integer

the 21st prime number and its reverse, 37, is the 12th prime number


number of non-Hamiltonian polyhedra with a minimum number of vertices


number of uniform polyhedra, excluding the infinite set

pentagonal pyramidal number


atomic number of the densest element, Osmium

number of books in the Catholic bible


largest integer that cannot be written as a sum of distinct numbers whose reciprocals sum to 1

maximum number of given digits in a sudoku puzzle, for it to lack a unique solution.


smallest integer that can be written as the sum of four distinct squares in three ways


atomic number of Gold.

fortunate, Gaussian, happy, Higgs, Kynea, lucky, permutable, Pillai and regular prime


days needed by Phileas Fogg to travel around the world

number of columns in IBM punch cards


number of stable chemical elements

only non-trivial number which is the square of the sum of its digits

its inverse 181 repeats all 10 decimal digits in order


sixth nuclear magic number

humber of 6-hexes, shapes made by joining six regular hexagons


atomic number of Bismuth, the heaviest stable element

Sophie Germain and Eisenstein prime


number of 5-faces of a seven-dimensional hypercube

largest order of a permutation of 14 elements


centered triangular, square, 14-gonal and 8-gonal number

number of ways to tie a tie


can be written as 222 in base 6

number of metals in the periodic table

happy number


sum of the squares of the first four primes

sum of the divisors of the first ten integers


only number known whose square has no isolated digits.

numbers of keys on a piano.

numbers of constellations in the sky

number of days in a year on Mercury


equals 81+92

11th Fibonacci number


number of degrees in a right angle

number of edges of a truncated Icosahedron

unitary perfect number and Pronic number

latitude of the North and South poles


smallest non-trivial cabtaxi number

sum of the squares of the first six integers.

smallest pseudoprime satisfying 3n=3modn.


number of faces of the snub dodecahedron

highest number of faces of any Archimedean solid

number of “atomic elements” in the Look-and-say sequence

atomic number of Uranium.


can be written as 333 in base 5

Mertens function returns 0 for 93

sum of the integers from 13 to 18

number of sets with 8 integers so that each is a proper divisor of the product of the others plus 1


number of graphic non-whitespace characters in ASCII

smith number, since the sum of its digits equals the sum of its prime factors

94! – 1 is prime


smallest non-interesting number


untouchable number, since it cannot be expressed as the sum of some of the proper divisors of another integer


the decimal expansion of its reciprocal starts with the powers of two: 198=0.010204081632


boiling point of water in Celsius

denoted by the letter C in Roman numerals

sum of the first four cubes: 100=13+23+33+43


number of heptominoes (one of which contains a whole)

sacred number in the Dharmic Religions


magic constant of the smallest magic square composed only of prime numbers and 1

magic constant of a 6×6 magic square


A113 was the Pixar animation classroom at the California Institute of the Arts


smallest number which can be written as the sum of three integers in four different ways, so that the product of every triple is the same: 118 = 14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 + 72


smallest number that is the product of 7 prime factors (27)

largest number that is not the sum of distinct squares

power of 2, all of whose digits are powers of 2 (unknown if there are any other numbers with this property)


smallest number that is the sum of all 2-digit numbers made from its digits 132=13+32+21+31+23+12

sixth Catalan number


one of just four numbers to be the sum of the factorials of its digits: 145 = 1! + 4! + 5!


sum of the cubes of its integers: 153=13+53+33

sum of the first 5 factorials and the first 17 natural numbers


the boiling point of water in Fahrenheit


largest number that can be represented using an 8-bit unsigned integer


smallest base 2 Fermat pseudoprime


average number of days in a year, consisting of 365 days, 5 hours 48 minutes and 46.08 seconds


number of octominoes


the A380 is the world’s largest passenger aircraft.


third perfect number


denoted by the letter D in Roman numerals


number of the beast


smallest palindromic square whose square root 26 is not palindromic


equals 6!

can be written as the product of consecutive integers in two different ways: 720 = 10 × 9 × 8 = 6 × 5 × 4 × 3 × 2


the Boeing 747 aircraft is called the “Queen of the Skies”


sacred number in Muslim Abjad numerology


second weird number, after 70

1 000

equals 103 in any number base

1 001

is the product of three consecutive primes 7 × 11 × 13

1 089

1089 × 9 = 9801, the reverse of its digits

the result if you take any 3-digit number, subtract its reversal, and the difference to its own reversal (for example, 752257=495 and 495+594=1089)

1 138

title of George Lucas’ first film: THX 1138

1 285

number of 9-ominoes

1 540

one of just 5 numbers that is both triangular and tetrahedral

1 729

Hardy–Ramanujan number and second taxicab number

smallest positive integer that can be written as the sum of two positive cubes in two different ways: 1729=123+13=103+93

2 021

current year

2 240

number of pounds in an (English) ton

2 310

smallest number with 5 different prime factors, since 2310=2×3×5×7×11

2 592

equals 25×92 and is the only number with such a pattern

3 003

smallest number to appear 8 times in Pascal’s triangle (there is no other number less than 2^23 appearing 8 times)

3 435

Munchausen Number: a number that is the sum of each digit raised to itself. The only non-trivial example is 3435=33+44+33+55

3 600

number of seconds in one hour

3 999

largest number that can be written in Roman numerals, as MMMCMXCIX

4 030

third weird number

4 900

the only square pyramidal number that is also a square: 4900=702=12+22+32++242

5 040

mentioned by Plato as the ideal number of people living in a city, due to its high number of factors

5 913

sum of the first 7 factorials: 5913 = 1! + 2! + 3! + … + 6! + 7!

6 174

Kaprekar’s constant: take any four-digit number, rearrange its digits to make the largest and smallest possible number, subtract these numbers and repeat. Eventually you will arrive at 6174!

8 000

equals 203, but is also the sum of four consecutive cubes: 113+123+133+143

8 128

fourth perfect number

9 240

has 64 divisors

12 758

largest number that cannot be represented as the sum of distinct cubes

17 163

largest number that is not the sum of the squares of distinct primes

40 320

equals 8 factorial

40 585

sum of the factorials of its digits: 40,585 = 4! + 0! + 5! + 8! + 5!

40 755

first number, other than 1, that is simultaneously triangular, pentagonal and hexagonal

142 857

smallest base 10 cyclic number: multiplied by 2, 3, 4, 5 or 6, it simply rotates is decimal digits

recurring decimal expansion of 17

196 560

the number of spheres touching any other sphere in a 24-dimensional Leech lattice

666 666

area of the Pythagorean triangle with sides 693, 1924 and 2045

2 598 960

number of distinct, 5-card poker hands

4 729 494

a coefficient in Archimedes’ cattle problem

31 557 600

number of seconds in a Julian astronomical year (365.25 days)

33 550 336

5th perfect number

275 305 224

the number of distinct magic squares of order 5 (excluding rotations and reflections)

299 792 458

speed of light in vacuum

739 391 133

the largest prime number (in base 10), that you can truncate to produce a sequence of smaller primes: 7, 73, 739, …

3 816 547 290

the only 10-digit number such that, for every k between 1 and 10, its first k digits give a number that is a multiple of k

8 589 869 056

6th perfect number

9 814 072 356

largest perfect power that contains no repeated digits in base ten

137 438 691 328

7th perfect number

158 753 389 900

the probability of receiving a complete suit in the game of Bridge

262 537 412 640 768 743.999999999


a transcendental number which is an almost integer

2 305 843 008 139 952 128

8th perfect number, discovered by Euler in 1772

18 446 744 073 709 551 615

equals 2641

the number of grains of rice required if you place one grain on the first square of a chess board, two on the second, 4 on the third, 8 on the fourth, and so on.

357 686 312 646 216 567 629 137

the largest prime number (in base 10), that you can truncate from its beginning to produce a sequence of smaller primes: 7, 37, 137, 9137, …

1 000 000 000 000 066 600 000 000 000 001

Belphegor’s Prime

666 with 13 zeroes either side