Zoo of Numbers: Numbers 201 to 684

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 centered icosagonal number, and a Kaprekar constant in base 4.
201n + 2 are prime, for n = 14, 15, and 16. - Patterson
201 x 627 = 126027
(the product has the same digits as its factors)

= 3 x 67

is the sixth Bell number.

is the largest number that cannot be written as the sum of distinct primes of the form 6n+1.

= 2³ + 2³ + 4³ + 5³ (sum of cubes)

is the smallest number whose English name contains all five vowels exactly once (twO hUndrEd And sIx).

= 1³ + 2³ + 2³ + 4³ + 5³ (sum of consecutive cubes)

is the smallest possible sum of primes which are formed using each of the digits 1 through 9:
207 = 89 + 61 + 43 + 7 + 5 + 2 - Gupta
curiously, the first half of the digits of 207⁴ are a permutation of the last half of the digits: 1836036801.

= 3² · 23
= 2² + 3² + 5² + 13²

CIVIC is the anagram of the Roman numeral CCVII (207).

is a tetranacci number.

= 2⁴ x 13
= 2² + 3² + 5² + 7² + 11² - Sladcik
= 2³ + 2³ + 4³ + 4³ + 4³

is a Harshad number and the smallest quasi-Carmichael number in base 9.
the curve 4²x² - y² = 209 contains the 'prime points' (3, 13), (5, 29), (7, 43), and (13, 83) - Buddenhagen

= 1⁶ + 2⁵ + 3⁴ + 4³ + 5² + 6¹ - posted by Jim O'Donohoe
= 3³ + 3³ + 3³ + 4³ + 4³
= 11 · 19

is a Chen prime.

= 3⁵ - 2⁵
= 67 + 71 + 73 (sum of consecutive primes)

is the lowest even 3-digit integer, ABC, such that (ABC)/(A x B x B) is also prime.
curiously, 4 out of 5 digits of 212² are the same (44944).
212° Fahrenheit = 100° Celsius (the boiling point of water)

= 6³ + (- 1)³ = 6³ - 1³ (sum and difference of cubes) - G. Sarcone
= 555 in base 6

is the smallest cube which can be written as the sum of 3 cubes: 3³ + 4³ + 5³
= 216 = 6¹⁺²

Is the 'constant' of the smallest multiplication magic square:

is a Gaglütz number: 217 modulo 712 = 217 - G. Sarcone
is a centered hexagonal number, and a Kaprekar constant in base 2.

= 1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 + 100 (sum of the factors of the number 100)
= 1³ + 6³ = (- 8)³ + 9³ (sum of cubes)

is a nontotient number, and the number of 4-vertex digraphs.

= 7³ + 13³ (sum of cubes)

is a semiprime number, and the number of space groups, not including mirror images.

= 3³ + 4³ + 4³ + 4³

is the smallest amicable number.

is the number of 7-vertex Hamiltonian planar graphs.

221b, Baker Street, London, England is the address of Sherlock Holmes.

= 10² + 11²
= 13 x 17
is the sum of 5 consecutive primes: 37 + 41 + 43 + 47 + 53
is also the sum of 9 consecutive primes:
11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41

is the smallest prime which will not remain prime if just one of its digits is changed.

= 71 + 73 + 79 (sum of consecutive primes)
= 19 + 23 + 29 + 31 + 37 + 41 + 43

is a prime number, and the number of 8-edge connected planar graphs.

= 2³ + 3³ + 4³ + 4³ + 4³

is the number of space groups, describing all possible crystal symmetries.

= 6² + 7² + 8² + 9² (consecutive squares)

is a centered triangular number (= [3xn² + 3xn + 2] / 2, n = 12).
is the smallest integer whose first two digits are different primes such that their sum equals the 3^rd digit.
is the number of trees with 11 vertices.

= 73 + 79 + 83 (sum of consecutive primes)

235 lunar months made up almost exactly 19 solar years (Metonic cycle: 19 tropical years ≈ 235 synodic months ≈ 6939.6... days )

the product of the digits of 236 is the reverse of the sum of its prime factors:
2 x 3 x 6 = 36 and 2 + 2 + 59 = 63

= 2² x 59
= 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 (sum of first consecutive primes)

is the largest number that cannot be written as a sum of 8 or fewer cubes.

This prime number appears in one of the earliest known geometrically converging formulas for computing :
/4 = 4 arctan(1/5) - arctan(1/239)

is the smallest number with 20 divisors.
n^x - n^x-4 is divisible by 240 if x > 7.
The maximum number of divisors for any integer less than 1,000,000 is 240. Only 5 numbers less than 1,000,000 have 240 divisors: 720720, 831600, 942480, 982800, 997920.

A cholesterol level of 240 and above is considered high risk.

is a prime number.

= (2⁸ + 4⁸ + 1⁸) / (2⁴ + 4⁴ + 1⁴)

is a nontotient, and the smallest number (except 2) that can be written as the sum of 2 squares or the sum of 2 fifth powers.

= 1³ + 3³ + 6³ (sum of cubes)

= ₉C₂ + ₉C₄ + ₉C₆

= 2 x 3 x 41
= 3³ + 3³ + 4³ + 4³ + 4³ (sum of cubes)

= 2⁸ - 2³

is the 5^th central binomial coefficient.

= (10 x 9 x 8 x 7 x 6)/(5 x 4 x 3 x 2 x 1)

252 x 252 = 441 x 144 (palindromic equality)

If you flip a coin 10 times in a row, there are exactly 252 ways in which it can turn out that you get exactly 5 heads and 5 tails.

is the smallest 8^th power (besides 1):
= 2⁸ = 4⁴ = 16²

= 1 0000 0000₂
= 100₁₆

= 2⁹ - 2⁸
= 4³ + 4³ + 4³ + 4³
= 3⁵ + 3² + 3¹ + 3⁰
256 = 2 x 5 + 6

is a Fermat prime.

_{= 2}2³_{+ 1}

= 1⁸ + 2⁸

is the constant of a 8x8 magic square.
= 2² x 5 x 13
= (1 + 2 + 3 + … + 62 + 63 + 64) / 8

is the number of days in the "Tzol k'in", the almanac cycle Mayans used for divination.

is is a prime number that is equal to the arithmetic mean of the nearest primes above and below: (269 + 257) / 2

= 43 + 47 + 53 + 59 + 61 (sum of consecutive primes)

seems to be the largest number whose square is 'undulating': 264² = 69696

n¹⁰ - 1 is divisible by 264 if n is prime and n > 3.

= 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 (sum of consecutive primes)

= 2³ + 2³ + 5³ + 5³

In nature, there are 266 different kinds of atoms.

is a deficient number, and the number of planar partitions of 12.

= 1³ + 2³ + 2³ + 5³ + 5³
= 2³ + 2³ + 2³ + 3³ + 6³ (sum of cubes)

seems to be the smallest number whose product of digits is 6 times the sum of its digits.

= 131 + 137 (sum of consecutive primes)

is a prime number, and twin prime with 271.

= 83 + 89 + 97 (sum of consecutive primes)

is a harmonic divisor number.
270 - 1 and 270 + 1 are primes.
10! has 270 divisors.

is a prime and a centered hexagonal number.
is the smallest prime p such that (p - 1) and (p + 1) are both divisible by a cube.

= 10³ - 9³ (difference of cubes)

is a prime and a Perrin number.

= 3³ + 5³ + 5³ (sum of cubes)
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... + 1/271 + 1/277 > 2

is a nontotient.

= 2 x 139

is a prime number.

= 29 + 31 + 37 + 41 + 43 + 47 + 53 (sum of consecutive primes)

is a tetrahedral number.
is the number of rooted trees with 9 vertices.

= C₁₃³ = C₁₃¹⁰

seems to be the smallest non-palindrome and non-square that when multiplied by its reverse gives a square:
288 x 882 = 504²

is the product of three distinct prime numbers:
= 2 x 5 x 29

= 3 x 97

is a prime number.

= 11 + 101 + 181 (sum of the first 3 tetradic primes)
= 2³ + 2³ + 3³ + 5³ + 5³ (sum of cubes)

is a self-number.

= 13 x 23

is a happy number, and a 6-hyperperfect number.

= 7 x 43
= 97 + 101 + 103 (sum of consecutive primes)

"I have some eggs in my basket. If I count them in 2s, in 3s, in 4s, in 5s, or in 6s I have always 1 egg left over. But if I count them in 7s there are no remainder. How many eggs are in my basket?" Solution: 301.

is a lucky number.

its cubic power is a concatenation of other cubes:
303³ = 27818127 (27 8 1 8 1 27)

relationships between 304 and 27:
304 = 3 x 10 x 10 + 4 and 27 = 3 + 10 + 10 + 4
304 = 2⁴ x 19 and 27 = 2 x 4 + 19

= 41 + 43 + 47 + 53 + 59 + 61 (sum of consecutive primes)

is the 5^th 'hexagonal prism' number. An hexagonal prism number is of the form (n + 1)(3n² + 3n + 1).

305² = 207² + 224² = 136² + 273² (Pythagorean triple)

= 5 x 61

is a prime, and a non-palindrome having a palindromic square:
307² = 94249

is a Harshad number, and a heptagonal pyramidal number.

= 151 + 157 (sum of consecutive primes)

is a permutable prime.

= 12² + 13²
is a palindromic prime.
is the only 3-digit number being a palindromic prime in base 10 and in base 2 (100111001) as well. - Larsen

3443 / 313 =11
37873 / 313 = 121
(all terms are palindromic!)

10³¹³ + 313 is prime

is the licence plate number on Donald Duck's car.

represents the number of unlabeled partially ordered sets of 6 elements.

= 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 (sum of consecutive primes)

cannot be represented as the sum of fewer than 19 fourth powers: 3 x 3⁴ + 4 x 2⁴ + 12 x 1⁴
is the smallest number with the property that the partition with the largest product does not have the largest number of parts. -Selfridge

= 103 + 107 + 109 (number of consecutive primes)

is the maximum determinant of a 10 by 10 matrix of zeros and ones.

320!+1 is prime.

= 2⁶ · 5 = 2⁵ · (2·5)

is the 12^th Lucas number.

The emblem of the secret society 'Skull and Bones' consists of a skull and crossbones, along with the number 322. The Order of Skull and Bones is based at Yale University, in New Haven, Connecticut.

is the product of twin primes (twin primes are pairs of primes of the form p, p+2): 323 = 17 x 19

= 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 (sum of consecutive primes)

= 73 + 79 + 83 + 89 (sum of consecutive primes)
324 = 3² x 4

324! - 1 is prime...

is a prime number.
= 11² - 10²
= 59 + 61 + 67 + 71 + 73 (sum of consecutive primes)

is the number of 7-hexes.
is a half-evil number: 666/2
= 3² x 37
= 370 - 37

336 x 951 = 319530
(the product has the same digits as its factors)

= 2⁴ x 3 x 7

= ₈P₃

= 6 x 7 x 8

is a permutable prime with 373 and 733.

_{= 2}2³ _{+ 3}2² _{- Kulsha}

is the smallest pseudoprime in base 2.
is a star number.
= 11 x 31
= 37 + 41 + 43 + 47 + 53 + 59 + 61 (sum of consecutive primes)
= 4⁰ + 4¹ + 4² + 4³ + 4⁴ (sum of four powers)

= 3 x 5 x 23

is a triangular number and the smallest number n where n, n+1, and n+2 are all products of 4 or more primes.

are the different arrangements of 9 non-attacking Queens on an 9 by 9 chessboard.

= 173 + 179 (sum of consecutive primes)

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