Almanac of Interesting Numbers

–1

largest negative integer

–0.083333333333333

112

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

0

zero

cardinality of the empty set

the additive identity

was represented using a shell in the ancient Mayan civilisation

0.5

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

0.577215664901532

Euler–Mascheroni constant γ

the limiting difference between the harmonic series and the natural logarithm

0.915965594177219

Catalan’s constant G

the result of the infinite series 112132+152172+

unknown whether irrational or not

1

the multiplicative identity

the only natural number that isn’t prime or composite

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

1.059463094359295

212

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

1.202056903159594

Apéry’s constant ζ3

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

appears in quantum electrodynamics and random spanning trees

1.259921049894873

23

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

1.414213562373095

2

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

1.618033988749894

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

1.732050807568877

3

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

2

smallest and only even prime number

Euler characteristic of a polyhedron homeomorphic to a sphere

2.236067977499789

5

length of the diagonal of a 1 × 2 rectangle

2.502907875095892

second Feigenbaum constant α

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

2.718281828459045

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

3

number of dimensions we can see

number of particles generations in the standard model

first Mersenne prime

3.141592653589793

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

4

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

4.669201609102990

first Feigenbaum constant δ

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

5

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

6

smallest perfect number

order of the smallest non-abelian group

number of distinct Quarks

number of regular polytopes

6.283185307179586

Tau τ=2π

circumference of a circle with radius 1

period of the sin, cos and tan functions

7

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

8

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”

9

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

10

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

11

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

12

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

13

number of different Archimedean solids

number of cards of a suit

unlucky number in Western culture

14

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

15

magic constant in a 3×3 magic square.

largest composite number with one one group of that order

triangular, hexagonal, pentatope and Bell number

16

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

17

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

18

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

19

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

number of cells in the only non-trivial magic hexagon

20

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

21

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

22

number of partitions of 8

number of Major Arcana cards in divinatory Tarot

pentagonal and a centred heptagonal number

23

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%

23.140692632779269

Gelfond’s constant eπ

24

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

25

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

automorphic, aspiring and non-sociable number

26

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

27

sum of the digits of its cube: 273=19683

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

28

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

29

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

Saturn requires over 29 years to orbit the Sun

30

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

31

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

32

ninth happy number

smallest nontrivial fifth power

32=11+22+33

sum of the totient functions of the first ten integers

freezing point of water at sea level in Fahrenheit

33

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

number of vertebrae in a human spine

sum of the first four factorials

34

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

35

number of different hexominoes

sum of the first five triangular numbers

highly cototient and tetrahedral number

36

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

37

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

38

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

39

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

40

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

41

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

42

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

43

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

44

number of derangements of 5 items

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

Tribonacci, happy and octahedral number

45

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

triangular, hexagonal and 16-gonal number

46

number of human Chromosomes

Erdős–Woods number

Wedderburn-Etherington number

47

largest number of cubes that cannot tile a cube

atomic number of silver

not a palindrome in any base b for 2b45

Carol number

48

double factorial of 6

number of Ptolemaic tessellations

highly composite number and the smallest number with 10 divisors

49

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

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

50

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

51

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

52

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

53

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

54

number of coloured squares on a Rubik’s cube

perfect round on a par 72 golf course

55

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

56

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

57

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.

57.295779513082320

number of degrees in 1 radian

58

number of commutative semigroups of order 4

sum of the first seven Prime numbers

59

number of stellations of an icosahedron

number of orthorhombic space groups

60

order of the smallest non-abelian simple group

base of the Babylonian number system

number of minutes in one hour and one degree

61

appears three times in the list of Fortunate numbers

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

62

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

written as 222 in base 5

63

number of partially ordered sets of 5 elements

64

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, …

65

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

66

sum of the first 22 integers

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

67

smallest number that is palindromic in bases 5 and 6

maximal number of letters in website domains

irregular and lucky and Pillai prime

68

truncated Tetrahedral Number

number of tetragonal space groups

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

69

its square and its cube together contain every digit exactly once

largest factorial on many calculators

70

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

71

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

72

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

73

least number of sixths powers needed to sum to every integer

the best number

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

a palindrome in binary:1001001

74

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

75

number of uniform polyhedra, excluding the infinite set

pentagonal pyramidal number

76

atomic number of the densest element, Osmium

number of books in the Catholic bible

77

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.

78

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

79

atomic number of Gold.

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

80

days needed by Phileas Fogg to travel around the world

number of columns in IBM punch cards

81

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

82

sixth nuclear magic number

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

83

atomic number of Bismuth, the heaviest stable element

Sophie Germain and Eisenstein prime

84

number of 5-faces of a seven-dimensional hypercube

largest order of a permutation of 14 elements

85

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

number of ways to tie a tie

86

can be written as 222 in base 6

number of metals in the periodic table

happy number

87

sum of the squares of the first four primes

sum of the divisors of the first ten integers

88

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

89

equals 81+92

11th Fibonacci number

90

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

91

smallest non-trivial cabtaxi number

sum of the squares of the first six integers.

smallest pseudoprime satisfying 3n=3modn.

92

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.

93

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

94

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

95

smallest non-interesting number

96

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

98

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

100

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

108

number of heptominoes (one of which contains a whole)

sacred number in the Dharmic Religions

111

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

magic constant of a 6×6 magic square

113

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

118

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

128

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)

132

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

145

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

153

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

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

212

the boiling point of water in Fahrenheit

255

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

341

smallest base 2 Fermat pseudoprime

365.2425

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

369

number of octominoes

380

the A380 is the world’s largest passenger aircraft.

496

third perfect number

500

denoted by the letter D in Roman numerals

666

number of the beast

676

smallest palindromic square whose square root 26 is not palindromic

720

equals 6!

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

747

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

786

sacred number in Muslim Abjad numerology

836

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

eπ163

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