Zoo of Numbers: Numbers 70 to 200

What's Special About This Number?

Fun facts on numbers (and their strange properties), both mathematical and cultural, tantalising problems and anecdotes. The number facts in this website are also available as features for print and electronic publishing.

is a Pell number and the smallest abundant number which is not the sum of some subset of its divisors.

= 7 + 8 + 9 + 10 + 11 + 12 + 13
= 2(6²- 1²) = 2(6 + 1)(6 -1)

70² = 1² + 2² + 3² + 4² + ... + 22² + 23² + 24²

Symbolism:
70 corresponds to the totality of an evolution, an evolutionary cycle being fully completed, according to Saint Augustin.
The corresponding Hebraic letter is 'ain, which corresponded to the sixteenth mystery of the Tarot: the house-God, symbol of fall, collapse, catastrophe - there is agreement with the idea of termination of cycle, according to R. Allendy.
In the Bible:
The 70 weeks prophecy of the book of Daniel. (Dn 9,24)
The 70 years of captivity of Babylonian of two of the twelve tributes of Israel. (2 Ch 36,21)
Ezekiel sees 70 ancients adoring the idols. (Ezk 8,11)

70 liters of water are needed to produce just 1 apple!

divides the sum of the antecedent primes, in fact:
2 + 3 + 5 + 7 + 11 + ... + 67 + 71 is divisible by 71.

71² = 7!+1
71³ = 357911 (consecutive primes)

is the smallest number that can be written as a sum of two primes, each ending with the digit 1, in two distinct ways:
72 = 11 + 61 = 31 + 41
is the largest number of spheres that can touch another sphere in a lattice packing in 6 dimensions.
Dividing 72 by the interest rate (x percent), you obtain the number of years you should wait to double your initial capital.

72 hours = 3 days

= 2³ · 3²
= 19² - 17²
= 11² - 7²
(difference of 2 consecutive squared primes)

is smallest prime whose digits are reversed in base 22 and the smallest number (besides 1) which is one less than the double of its reverse: 2 x 37 - 73 = 1
is a divisor of 10001.

73² = 48² + 55² (Pythagorean triple).

365 days in a year : 5 = 73

is the smallest number with the property that it and its neighbors can be written as a sum of 2 squares.
72 = 6² + 6²
73 = 3² + 8²
74 = 5² + 7²
- Found by Daniel Clemente from the Polytechnical University of Catalonia

is the number of orderings of 4 objects with ties allowed.
is a repfigit number...

If you take the "digit sum" 7+5=12 and then continue adding in twos in "fibonacci style" look what happens:
7 + 5 = 12
5 + 12 = 17
12 + 17 = 29
17 + 29 = 46
29 + 46 = 75 ...arriving back at 75!

75⁵ = 19⁵ + 43⁵ + 46⁵ + 47⁵ + 67⁵

75 liters of water are needed to produce 1 glass of beer (250 ml)!

is an automorphic number:
76² = 5776
76³ = 438976
76⁴ = 33362176 ...

= 8 + 13 + 21 + 34 (sum of consecutive Fibonacci numbers)
= 3² + 3² + 3² + 7² (sum of primes)

is the largest number that cannot be written as a sum of distinct numbers whose reciprocals add up to 1. For example:
2 + 6 + 8 + 10 + 12 + 40 = 78 and
1/2 + 1/6 + 1/8 + 1/10 + 1/12 + 1/40 = 1
is the sum of 3 squares:
4² + 5² + 6²
is the sum of the first 8 prime numbers:
2 + 3 + 5 + 7 + 11 + 13 + 17 + 19

= 1001 / (77 - 37 - 27) - By G. Sarcone
= 4x4 + 5x5 + 6x6
= 39² - 38²
= 9² - 2²

Any number having a abcabc numeral pattern is divisible by 77 (replace each letter with a digit. E.g. 123123)

Seventy-seven is the smallest number requiring five syllables in English.

French speaking people have two different names for seventy-seven:
'Soixante-dix-sept' (sixty and seventeen) and 'septante-sept' (Swiss French people).

During World War II in Sweden at the border with Norway, "77" (sjuttiosju) was used as a 'Shibboleth' (discriminatory password), because the tricky pronunciation in Swedish (approx. 'shütioohshü') made it easy to instantly discern whether the speaker was native Swedish, Norwegian, or German.

is a permutable prime, and the smallest prime whose sum of digits is a 4th power.
is a Pillai prime. A Pillai prime is a prime p for which there is an integer n > 0 such that the factorial of n is 1 less than a multiple of the prime, but the prime is not 1 more than a multiple of n, in short: n! ≡ -1 (mod p) but p ≠ 1 (mod n)

79 = 11 + 31 + 37 and
97 = 11 + 13 + 73 (sums of reversible primes)

= (7 x 9) + (7 + 9) (this also works for 19, 29, 39, 49, ...)

= 40² - 39² = 40 + 39

= 2⁷ - 7²
= 2² + 5² + 5² + 5² (sum of squares)

≈ 8.888²

x² - 79x + 1601, if you replace x in the previous expression with any number from 1 to 79, you always obtain a prime number!

is the product of 5 primes.
is the smallest number n where n and n+1 are both products of 4 or more primes (80 = 2 x 2 x 2 x 2 x 5 and 81 = 3 x 3 x 3 x 3).

= 4 scores

= n x 2^(n-1) = 5 x 2⁴

= 3 + 5 + 7 + 9 + 11 + 13 + 15 (sum of consecutive odd numbers)
= 7 + 73 = 13 + 67 = 19 + 61 = 37 + 43 (4 distinct sums of 2 primes)
= 2³ + 2³ + 4³ (sum of 3 positive cubes)

= 4² + 8² = 4² + 4³
= 9² - 1² = 292² - 44³

= 6! / 3²

80² = 4³ + 8³ + 12³ + 16³

The "80:20 rule" originated from Vilfredo Pareto, an Italian economist. He discovered a common phenomenon: about 80% of the wealth in most countries was controlled by a consistent minority -- about 20% of the people.

In French there are 3 ways to say 80:
'quatre-vingts', in France, and Quebec,
'huitante', in Switzerland,
and 'octante', in Belgium.

the sum of its digits is a square.
is the square of the sum of its digits.
is the 3rd lowest prime power of a prime power of a prime.

= 3⁴ = 9²
= 2⁵ + 7²
= 41² - 40² = 41 + 40

x² - 81x + 1681, if you replace x in the previous expression with any number from 1 to 80, you always obtain a prime number!

1/81 = 0.123456791234567912345679...
(The result is a repeating decimal containing every digit from 1-9 except for 8) - Posted by Ting Wang

is the number of zeroless squares that are composed of the digits 1 up to 9. - Beilier

= 11 + 31 + 41 (sum of the first 3 primes ending with 1)

it seems there are over 10⁸³ elementary particles with non-zero mass in the known universe.

is a tetrahedral number:
84 = 1 + 3 + 6 + 10 + 15 + 21 + 28 (sum of the first 7 triangular numbers)
is the highest order of a permutation on 14 elements.

= 41 + 43 (sum of twin primes)
= 4¹ + 4² + 4³
= 3² + 5² + 5² + 5² (sum of cubes)

1000 + 84 = 1084 (one thousand eighty-four) is the smallest natural number whose name contains the five vowels a, e, i, o, u in order.

is the largest known n for which 1²+2²+3²+...+n² = 1+2+3+...+m has a solution.
85/38 is a good approximation of 5.

= 2² + 9²
= 6² + 7² (sum of 2 consecutive squares)
= 4⁰ + 4¹ + 4² + 4³ (sum of powers of 4)
85² = 13² + 84² = 36² + 77²

is an happy number.
is the largest known n for which 2ⁿ contain NO zeros:
2⁸⁶ = 77,371,252,455,336,267,181,195,264 (26 digits).

= 222 in base 6.
= 20 + 21 + 22 + 23
= 3² + 4² + 5² + 6² (sum of consecutive squares)

used as a verb, to "eighty-six" means to "ignore" or "get rid of".

is the sum of the squares of the first 4 primes:
2² + 3² + 5² + 7²

= 3 x 29
= 12 + 13 + 14 + 15 + 16 +17
= 44 + 43 = 44² - 43²

By tradition, 87 is an unlucky score for cricketers (100-13).

is the only number known whose square has no isolated digits.
A piano has 88 keys.
= 23² - 21²
= 13² - 9²
It can be read the same upside down or when viewed in a mirror.

The 8th letter of the alphabet is 'H'. 8 two times signifies 'HH', shorthand for the Nazi greeting:"Heil Hitler". 88 is often found on hate group flyers, in both the greetings and closing comments of letters written by neo-Nazis, and in e-mail addresses.

is a prime and a Fibonacci number.
generates a cyclic number series where each term is the sum of the square digits of the previous number:
89 - 145 - 42 - 20 - 4 - 16 - 37 - 58 - 89 - ...

89² = 39² + 80² (Pythagorean triple)
= 8¹ + 9²
= 2³ + 3³ + 3³ + 3³

is the number of degrees in a right angle.

= 1 x 2 x 3² x 5
= 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 (sum of the first 9 consecutive even numbers)
= 7 + 11 + 13 + 17 + 19 + 23 (sum of consecutive primes)
= 2² + 3² + 4² + 5² + 6² (sum of consecutive squares)
= 9¹ + 9² = 10² - 10¹

Sturgeon's Law: "90% of everything is rubbish"

Some 'scientific' reviews state humans are "90% water" when in fact newborns have approx. 78%, 1-year-olds around 65%, adult men about 60%, and adult women around 55%. See mythical numbers.

In Italian pezzo da novanta (piece of ninety) means 'big boss' or 'big shot'. In Italy, when someone is or has been frightened by something unexpected, it is used to say la paura fa novanta! (the scare does ninety!).

is a triangular, an hexagonal and also a square pyramidal number.
is a semiprime and the smallest pseudoprime to base 3.

= 1 + 2 + 3 + ... + 11 + 12 + 13 (sum of consecutive numbers)
= 1² + 2² + 3² + 4² + 5² + 6² (sum of consecutive squares)

= 3³ + 4³
= 6³ - 5³

The traditional dimensions of the tatami mat (畳) are 91 x 182 cm.

In 2006, 91 per cent of all known executions took place in China, Iran, Pakistan, Iraq, Sudan and the USA.
Executions in 2006:
China: 1,010+; Iran: 177; Pakistan: 82; Iraq: 65+; Sudan: 65+; USA: 53; Saudi Arabia: 39+; Yemen: 30+; Vietnam: 14; Kuwait: 10+; Somalia: 7+; Singapore: 5+; Egypt: 4+; Jordan: 4+; Japan: 4; Malaysia: 4; Indonesia: 3+; Bahrain: 3; Mongolia: 3; North Korea: 2; Syria: 2; Uganda: 2; Bangladesh: 1; Botswana: 1; Equatorial Guinea: 1.
Note: '+' symbol indicates that the figure is a minimum one; the true figure may be higher due to state secrecy or a lack of available information.

= 3² + 3² + 5² + 7² (sum of squares)

is the number of different arrangements of 8 non-attacking Queens on an 8 by 8 chessboard (i.e. no two Queens should share the same row, column, or diagonal).

		92
		 736
		  5888
		   47104
		    376832
		     3014656
		      24117248
		       192937984
		9999999...

= 333 in base 5.
With just 9 straight cuts a potato can be divided into 93 pieces.

= 2² + 5² + 8² (sum of squares)
= 13 + 14 + 15 + 16 + 17 + 18

is a 17-gonal number and a Smith number (a Smith number is a composite number the sum of whose digits is the sum of the digits of its prime factors excluding 1, here: 9 + 4 = 2 + 4 + 7).

is the number of planar partitions of 10.

= 1² + 2² + 3² + 9² (sum of squares)
= 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15
= 5 + 7 + 11 + 13 + 17 + 19 + 23 (sum of consecutive primes)

is the smallest number that can be written as the difference of 2 squares in 4 ways:
= 25² - 23² = 14² - 10²
= 11² - 5² = 10² - 2²

is the largest two-digit prime whose cube is zeroless.
97/56 is a good approximation of 3.
curiously, the numbers 97, 907, 9007, 90007 and 900007 are all prime.

97² = 65² + 72² (Pythagorean triple)

= 49² - 48² = 49 + 48
= 29 + 31 + 37 (sum of consecutive primes)
= 4!4 + 4/4

0.097 is the average brain weight of a goldfish (in grams).

is the lowest number such that its first 5 multiples contain the digit 9: 098, 196, 294, 392, 490

= 1⁴ + 2⁴ + 3⁴

98.14072356 is 9.9066 squared, and it used the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 exactly once.

is a palindromic number and a Kaprekar number.

= 2³ + 3³ + 4³
= 4 + 5 + 6 + 7 + ... + 11 + 12 + 13 + 14
99/70 is a good approximation of 2.
99² = 9801 and 98 + 01 = 99

XCIX or IC?
Rules regarding Roman numerals often state that C cannot be preceded by I or V, only by X. Thus, one should represent the number 'ninety-nine' as XCIX, not as the 'shortcut' IC.
This 'problem' manifested in questions as to why 1999 was not written simply IMM or MIM as opposed to the virtually universal MCMXCIX.

Machoumearobilengmonoolemongametsoarobilengmonoolemong means '99' in the language of the Bassoutos tribe.

Riddle: What goes "99 bonk"?
Answer: A Centipede with a wooden leg!

is the smallest square which is also the sum of 4 consecutive cubes.

= 33 x 3 + 3/3
= 5 x 5 x 5 - 5 x 5
= (8 + (8 + 8)/8)(8 + (8 + 8)/8)

= 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 x 9
= (1 + 2 - 3 - 4)(5 - 6 - 7 - 8 - 9)
= 123 - 45 - 67 + 89
= 56 + 34 + 90/(82-71) [by G. Sarcone]
(in the operation above all digits from 0 to 9 were used once)

= 1³ + 2³ + 3³ + 4³ = (1 + 2 + 3 + 4)²
= 26² - 24² = 6² + 8²

= 47 + 53 (sum of consecutive primes)
= 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 (sum of consecutive primes)

The human brain is a busy place. We each have about 100 billion neurons. Each of the 100 billion neurons actively communicates with 1000 - 10,000 others.
Each of the 100 billion neurons can be 'on' or 'off', and can generate 2^{100,000,000,000} (or roughly 10^{30,000,000,000}) unique brain states. The number of the possible brain states is greater than the number of particles in the universe!

101² = 20² + 99² (Pythagorean triple)

is the number of partitions of 13.

is the lowest number with 3 different digits.

= 2 x 3 x 17

is the only odd prime number that divides the number of bones in a human body.

= 2² + 5² + 5² + 7²

is the smallest number other than 1 that is impossible to score with two darts.

= 13 x 2³ = (13 - 2³)³ - 13 - 2³

is the lowest known number of unit line segments that can exist in the plane, 4 touching at every vertex.

is the largest number n known such that n - 2^k is prime for k>1. - Paul Erdös

is the number of trees with 10 vertices.
The sum of the first 106 digits of Pi is prime. - Sellers

is the exponent of a Mersenne prime.

= 2³ + 2³ + 3³ + 4³

is the smallest positive integer requiring 6 syllables in English.

is 3 hyperfactorial.
can be written as the sum of a cube and a square (a³ + b²) in two ways (3³ + 9² and 2³ + 10²). This is the smallest number with this property.
Every angle of a pentagon measures 108°.
There are 108 different types of heptominoes.

= 36 x 3 = 27 x 4
= 0⁰ x 1¹ x 2² x 3³
= 3³ + 3³ + 3³ + 3³ = 6² + 6² + 6²
= 12 x 9 and 1 + 0 + 8 = 9
= 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16
= 3² + 5² + 5² + 7² (sum of square primes)
= 8 + 15 + 29 + 56 (sum of consecutive Tetranacci numbers)

is a significant number in Buddhism. It represents the number of temptations that human beings face, which tie them to the Wheel of samsara. There are many religious elements related to this number - a bell is rung 108 times to bring in the New Year in Japan.

is the number of beads in a Buddhist Japa mala (kind of rosary). They implement the following formula: 6 x 3 x 2 x 3 = 108
6 senses (sight, sound, smell, taste, touch, thought),
3 aspects of time (past, present, future),
2 conditions of heart (pure or impure),
3 possibilities of sentiment (like, dislike, indifference).

is the sum of the numbers 4, 8, 15, 16, 23 and 42 that frequently appear throughout the TV series 'LOST'.

The diameter of the sun is 108 times the diameter of the Earth.

is the smallest number which is palindromic in bases 5 and 9.
Using only the coins in circulation in the UK, you can pay the sum of 30 pence in 109 different ways.
The interior angle of a tetrahedron is 109° 28’.
Soap bubbles can only join each other at one or two angles: 109 or 120 degrees. No other angles are possible.

= 31 + 37 + 41 (sum of consecutive primes)
109² = 60² + 91² (Pythagorean triple)

is the smallest number that is the product of two different substrings.

= 5² + 6² + 7² (3 consecutive squares)
= 2 x 55 = 5 x 22
= 1x2 + 2x2 + 3x2 + ... + 8x2 + 9x2 + 10x2

is also known as "eleventy" according to the number naming system invented by J. R. R. Tolkien.

is the smallest possible magic constant of a 3 x 3 magic square of distinct primes.

is the ninth number that stays the same when written upside down.
is also the 'magic constant' of the n-Queens Problem for n = 6.

= 3 x 37
= 12 x 9 + 3
= 16 + 17 + 18 + 19 + 20 + 21

is an abundant number and a heptagonal number.
is the side of the smallest square that can be tiled with distinct integer-sided squares.

= 2⁴ x 7

is a permutable prime.
355/113 = 3.1415929... A good approximation of .

There are 13 consecutive divisible integers (non-primes) between 113 and 127.

For Tintinologists
113, and not 57b as depicted in the comic book 'the Calculus Affair' by Hergé, is the number of the street where the house of the professor Alfredo Topolino is still located in Nyon. Actually, the current full address is: 113, route de Saint-Cergue, 1260 Nyon, Switzerland.

appears in the Padovan sequence, preceded by the terms 49, 65, 86 (it is the sum of the first two of these).

= 222 in base 7.
114 = 19 x 6 and 1 + 1 + 4 = 6

The Muslim holy book the Quran is divided into 114 suras or chapters.

is a heptagonal pyramidal number and a lucky number.
is the number of 8-vertex rooted trees.

= 23 x (2 + 3)
= 5! - 5

is the lowest value of the longest edge in a Heronian Tetrahedron.
117² +/- 2 are primes.

= 11³ - 2³ = 5³ - 2³

is the smallest 3-digit number having two prime digit permutations: 191 and 911.

= 17 + 19 + 23 + 29 + 31 (sum of consecutive primes)

is a triangular number.
is the smallest number to appear 6 times in Pascal's triangle.
= 5!

120 liters of water are needed to produce 1 glass of wine (125 ml)!

is the only square known of the form 1+p+p³+p³+p⁴, where p is prime (121 = 3⁰ + 3¹ + 3² + 3³ + 3⁴).

121 = 38 + 83 (palindromic sum)

= 5! + 1
= 37 + 41 + 43 (sum of 3 consecutive primes)
= 11²
= 1 + 3 + 5 + 7 + 9 + 11 + ... + 19 + 21 (sum of consecutive odd numbers)

= 29 + 30 + 31 + 32
= 115 + (1 + 1 + 5)

Palindromic curiosities with 122:
122 x 122 = 14884 and 48841 = 221 x 221
122 x 213 = 25986 and 68952 = 312 x 221

is the 10^th Lucas number.

= 18 + 19 + 20 + 21 + 22 + 23

123 x 2¹²³ – 1 = 1.30796035 x 10³⁹ =
= 1307960347 8523572189 3734614731 5859062783 =
= prime number

123 modulo 12 = 3 -- G. Sarcone

is the symbol of the Holy Trinity: Father, Son and Holy Spirit.
The number 123 is used 3 times in the Bible.

is an untouchable number.
is the smallest number such that its first 3 multiples contain the digit 2.

is the smallest number which is both a sum of distinct prime squares and a prime cube, and is the only number known that contains all its proper divisors as proper substrings.

is a Mersenne prime number.
= 127 = 2⁷ - 1

is the sum of seven powers of 2:
= 2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ + 2⁶
You can arrange the number 127 as a pattern of concentric hexagons.

is a noncototient.
is the number of functions from 6 unlabeled points to themselves.

= 1² + 2² + 5² + 10² (sum of the squares of the first 4 divisors)

is prime, and its reversal (331) is also prime.
is the smallest integer n for which the sum of the proper divisors of n divides phi(n).

_{= 2}2²_{+ (2 x 3)}2 _{+ 3}2² _{- G. Sarcone}

= 2³ + 5³

divides 11,111,111.
is the maximum number of 7^th powers that are needed to sum to any arbitrarily large number.
is the largest prime factor of 123,456,787,654,321.
is the average brain weight of a baboon (in grams).

= 10001 / 73

is the smallest possible product of 3 primes, one of which is the concatenation of the other two (2 x 3 x 23).
is the sum of four consecutive primes: 29+31+37+41

138 + 138/2 = 207 and 0.207207207... x 666 = 138

'Untrioctium' is an unsynthesized chemical element with atomic number 138 and symbol Uto.

is considered as an occult number and one of its factors (23) is also an occult number!

is the smallest harmonic divisor number:
12 / (1/1 + 1/2 + 1/4 + 1/5 + 1/7 + 1/10 + 1/14 + 1/20 + 1/28 + 1/35 + 1/70 + 1/140) = 5

= 1² + 2² + 3² + 4² + 5² + 6² + 7²

140 liters of water are needed to produce 1 cup of coffee (125 ml)!

is a Cullen number (n x 2ⁿ + 1 is a Cullen prime when n = 141).

= 2³ + 2³ + 5³ (sum of cubes)

is the smallest quasi-Carmichael number in base 8.

= 11 x 13 (product of twin primes)
= 1001 / 7
= 2² + 3² + 3² + 11²
= - 3⁴ - 4⁴ - 5⁴ - 6⁴ + 7⁴
= 43 + 47 + 53 = 11 + 13 + 17 + 19 + 23 + 29 + 31 (sum of consecutive primes)

is the only square in the Fibonacci sequence (apart from 1). Moreover, it is the 12th term of the Fibonacci sequence.

Reversing 12 and 144 in the equality 12² = 144 gives 441 = 21².

144⁵ = 27⁵ + 84⁵ + 110⁵ + 135⁵

= 71 + 73 (sum of consecutive primes)
= 3! x 4! = 4! + 5!
= 4² + 2 x 8²
= 6² + 3 x 6²

One of the largest numbers, that is not a power of 10, that has a specific name: 'gross' (= dozen dozen).

is a centered square number and also a pentagonal number.
is a product of 2 distinct primes: 5 x 29

= 1! + 4! + 5!
= 4³ + 3⁴
= 12² + 1² = 8² + 9² (sum of squares)
= (1² + 2²)(2² + 5²)
= 3³ + 3³ + 3³ + 4³ (sum of cubes)

is an octahedral number. An octahedral number is of the form (2n³ + n)/3.

= 222 in base 8
= 5² + 11²
= (1² + 1²)(3² + 8²)

146² + 1 is prime

is an odd nonprime number divisible by exactly 3 primes:
= 3 x 7 x 7
is the number of sided 6-hexes.

is an heptagonal number.
is the number of 6-vertex perfect graphs.

is the concatenation of the first 3 positive squares.

= 6² + 7² + 8² (sum of consecutive squares)
= 2³ + 2³ + 2³ + 5³ (sum of cubes)
= 23 + 43 + 83 (sum of 3 primes which are not sum of two squares)</