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204 = 17*12, the product of the numerator and denominator of 17/12, which is an approximation to the square root of 2 formed by the Pell numbers and another similar sequence. The first few numbers like this are: 1, 6, 35, 204, 1189, 6930, 40391, 235416, 1372105, 7997214, ... (Sloane's integer sequence A001109). The sequence follows the pattern:
An = 6 An-1 - An-2
(for example, 204=6×35-6 and 1189=6×204-35).
A really nifty property of these numbers is that their squares are also triangular — see 41616 for more.
Approximate mass ratio between an electron an a muon. The muon is about 207 times heavier, as a result when it binds with a nucleus its charge distribution is about 207 smaller. This means that a muon bound with a proton forms an atom that is chemically similar to hydrogen but with 207 times smaller diameter. See also 137.03599....
210 is a primorial: the product of the first N primes: 210=2×3×5×7. The primorials are: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, ... (Sloane's A002110).
210 is a 3-d oblong number, the product 5×6×7. Since it is also 14×15, it is another example of a number that is the product of consecutive integers in two different ways (see also 120, 720, 5040).
For a while, 2 is the most common gap between consecutive prime numbers. Then (starting at about 100, and permanently after 1000), gaps of 6 occur most often. Up at around 1035, it shifts to 30. There is a conjecture that above about 10425 the most common prime gap is 210.
Base 210 is also a record-setter for tests for divisibility by primes. See 66.
216 is a cube, 63, and is the sum of three consecutive cubes72: 33 + 43 + 53 = 63. See also 25, 143, 8000 and 31858749840007945920321.
8 pieces are required to construct a puzzle that can be assembled as
a 6×6×6 cube and reassembled into the three smaller cubes. The 6
"slices" of the puzzle would appear as follows, with letters a
through h representing the eight puzzle pieces:
f f f f f h f f f f f h f f b b b b
f f f f f e f f f f f e f f b b b b
f f f f f e f f f f f e f f f b b b
g g g e e e g g g e e e g g g b b b
g g g e e e g g g e e e g g g e e e
g g g e e e g g g e e e g g g e e e
slice-1 slice-2 slice-3
.
f f b b b b a a b b b b a a b b b b
f f b b b b a a b b b b a a b b b b
c c b b b b c c b b b b c c b b b b
c c b b b b c c b b b b c c b b b b
c c c e e e c c c e e e c c c d d d
c c c e e e c c c e e e c c c d d d
slice-4 slice-5 slice-6
216 = 8×27 = 23×33, and had spiritual significance in ancient times because it was the product of the first two cubes (they didn't count 13 as a cube).
It is also 6×6×6, and is the value of Φ(666), the number of integers less than and relatively prime to 666.
In the 1998 movie π (a.k.a. "Pi the Movie"), the Qabbalistic Jews are searching for a sequence of 216 Hebrew letters, or a 216-digit number (which is the same thing in ancient Hebrew, since the letters of the Hebrew alphabet are used to represent the digits of numbers).
220 and 284 form an amicable pair, an idea that goes back to the time of Pythagoras. If you add the factors of 220 you get 284 and vice versa. The search for amicable numbers is closely related to the classical question of finding perfect numbers.
231 appears as a denominator in the "greedy" Egyptian fraction for 3/7: 3/7 = 1/3 + 1/11 + 1/231.
To the best of our knowledge, the ancient Egyptians had no general notation for an arbitrary fraction A/B. They had a special symbol for 2/3, and any other fractions were expressed as a sum of "unit fractions", fractions of unity, such as 1/3 and 1/7. For example, instead of 3/4 they wrote 1/2+1/4, and instead of 5/6 they wrote 1/2+1/3. By convention, they always expressed it as a sum of different unit fractions (for example, 2/7 could not be written 1/7+1/7, but 1/4+1/28 or 1/5+1/20+1/28 were OK), and the fractions were always listed largest-first (that is, smallest denominators first).
It is not entirely known why they did it this way. It could have been for practical reasons in performing physical divisions. For example, imagine dividing 2 equal bushels of grain among 7 people. Each person should get 2/7 of a bushel. The simple approach would be to divide each into 7 equal parts (give each person 1/7 plus 1/7). But a fair division into 7 parts is difficult; 4 is much easier (use a balance, or the "I split, you choose" method, to divide in half; then repeat). So, divide each bushel into 4 equal parts, and give 1/4 bushel to each person; the remaining 1/4 bushel then is divided into 7, which is much easier than the original task. 2/7 = 1/4 + 1/28.
It is possible to express any fraction as a sum of unit fractions, and usually in more than one different way. There are several different algorithms for converting a fraction like 3/7 into an Egyptian fraction sum. One, called the "greedy" algorithm, works by subtracting the largest possible unit fraction each time and repeating:
Given: Fraction X/Y, where X and Y are positive integers and X
2: 1/A is the first (or next) term in the Egyptian fraction.
3: Compute X/Y - 1/A and reduce to lowest terms. The result is
the new X/Y. If nonzero, go back to step 1 and repeat.
This algorithm always produces an answer with a finite number of terms, but the denominators sometimes get really large. Starting with 3/7 we get:
3/7 = 1/3 + 2/21
2/21 = 1/11 + 1/231
therefore 3/7 = 1/3 + 1/11 + 1/231
This prime number appears in one of the earliest known geometrically converging formulas for computing π:
π/4 = 4 arctan(1/5) - arctan(1/239)
= SUM [ (-2n)/(2n+1)(52n+1) - (-1n)/(2n+1)(2392n+1) ]
This formula works because of this special relationship between 5 and 239 through their squares:
2 × 134 = 2392 + 1 = 57122
and
2 × 13 = 52 + 1 = 26
and
arctan(x) + arctan(y) = arctan( (x + y) / (1 - x y) )
This relationship makes it possible to show that a geometric construction of 4 right triangles in the proportion 1 :: 1/5 :: (√26)/5 and one triangle in the proportion 1 :: 1/239 :: (√57122)/239 can be used to produce a triangle in the proportion 1 :: 1 :: √2. Also related to this is the fact that 239/169 = 239/132 is a good close approximation to the square root of 2.
You might wonder, why do all that when you could compute π/4 directly from the arctangent of 1:
π/4 = arctan(1)
= SUM [ (-1n)/(2n+1) ]
= 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
The reason is that this series converges too slowly to be of any use. The 52n+1 in the denominator in the series above makes it converge quickly enough so that you get about 1.398 = log10(52) digits of π for each term you evaluate.
239 is the smallest factor of 9999999 that is not also a factor of some smaller string of 9's, and therefore 239 is the smallest number whose reciprocal has a 7-digit repeating pattern: 1/239=0.00418410041841004184... See also 27 and 757.
Here are two rather obscure properties of 239:
239 is the largest number that cannot be expressed as the sum of 8 (or fewer) cubes (it requires 9: 239 = 53 + 33 + 33 + 33 + 23 + 23 + 23 + 23 + 13). The only other such number is 23. 239 also requires the maximum number of terms to be expressed as a sum of squares (4 squares, 239 = 152 + 32 + 22 + 12) or of 4th powers (19 4th powers). It does not require the maximum number of 5th powers.
Define the "sum of prime factors" sopf(N) to be the sum of each of the prime factors of N, counting a prime more than once if N is a multiple of its square, cube, etc. So, for example, sopf(42) = 2 + 3 + 7 = 12, and sopf(27) = 3 + 3 + 3 = 9. sopf(N) = N only if N is prime, or if N is 1 or 4. Now consider the sum N + sopf(N). As N increases, this value increases irregularly. Some values never occur — for example, there is no N such that N + sopf(N) equals 12 — and other values occur more often — for example, there are two 14's and two 23's. The value 239 occurs often enough that if you add the sopf(N)'s for all N's that have N+sopf(N)=239, you get a sum greater than 239. 239 is the first number for which this occurs (the next few are 1439, 2159, 4319).
(Personal: 239 is another street number where I have lived. The famous HAKMEM72 file was MIT AI memo number 239.)
240 is a "Fibonacci factorial", the product of the first 6 Fibonacci numbers: 1×1×2×3×5×8 = 240. See also 158.
243 = 35 = 9×27, and has a few cool properties. It is a perfect totient number, along with all other powers of 3. There is more about these numbers on a separate page here.
If we divide 53 by 35, we get a 27-digit repeating decimal with the digits 514403292181069958847736625. These 27 digits taken as an integer have the 370 property: the average of all possible permutations of its digits is equal to itself.
247 is an Eulerian number. The Eulerian numbers, given by Sloane's A000295, begin: 0, 1, 4, 11, 26, 57, 120, 247, 502, 1013, 2036, ... and are simply the values of 2n-n-1. These numbers usually manifest themselves as a permutation count. In this figure, each of the permutations has exactly one place where a larger number follows a smaller number (called an "ascent"):
A[2] = 1 12
A[3] = 4 132 213 231 312
A[4] = 11 1432 2143 2431 3142 3214 3241 3421 4132 4213 4231 4312
The Eulerian numbers show up as the first non-trivial diagonal of a number triangle (called Euler's triangle) that is like Pascal's triangle, but with rows that add up to successive factorials:
To compute numbers in this triangle, you add the two numbers above it, each multiplied by the index number of the diagonal shared by that number. For example, consider the first 26 in the 4th row. The numbers 1 and 11 are above it. The 26 and the 11 share the 2nd diagonal, counting from the left, so we take 11×2. The 26 and the 1 are in the 4th diagonal, counting from the right, so we take 1×4. This gives 11×2 + 1×4, which is 26. For another example, 57×5 + 302×3 = 1191: the multipliers 5 and 3 are used because the number to be computed is in the 5th and 3rd diagonals.
The first few diagonals of the Euler triangle are A000295, A000460, and A000498, and the largest number in each row is given by A006551. All of these count permutations, but with different requirements for how many "ascents" there are.
251 is the first of a set of 4 consecutive primes that are spaced an equal distance apart: 251, 257, 263, and 269 are all prime, there are no primes in between, and the spacing between them is 6. 251 is the lowest number with this property; the next is 1741. See also 47, 9843019, 121174811 and 19252884016114523644357039386451.
252 = 10×9×8×7×6 / (5×4×3×2×1)
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 (some examples are HTHTTTTHHH, THTHTHTHTH, HHHHHTTTTT, TTHHHHTHTT). Note the rather elegant way of expressing this using the numbers 1 through 10. This type of expression can be used to calculate any of the numbers down the center of Pascal's triangle. See also 2520.
This is 28, and is the number of possible values that can be stored in a byte, the most common unit of measurement for computer memory quantities. The number of values results from the fact that there are eight bits in a byte, and each bit can have two values (0 or 1). (Yes, during one period of history (16th century and earlier) there were also eight "bits" in a Spanish dollar, a fact that is related to the phrase "pieces of eight", and "two bits" (as in "shave and a haircut — two bits", slang for "25 cents"). The Spanish dollar was a gold coin with a value of eight reale, and was sometimes actually cut into eight wedge-shaped pieces to make change.)
Eight bits became the standard computer memory quantity because it is also a power of 2 (other smaller "bytes" were used in the early days when memory was expensive, but having the number of bits be a power of two makes many operations more efficient.) The next smaller power-of-2 size would be 4 bits per "byte", but that only allows for 16 = 24 possible byte values. The need to store and manipulate text, including a full alphabet (26 characters in the country where most of the computers were being developed) plus digits and many punctuation symbols meant that a byte that takes at least 100 or so values was necessary.
If computers had been developed in one of the many countries that have over 256 writing symbols (those that use the Korean, Ethiopic, Japanese, Chinese, and Vietnamese writing systems, among others) they might well have settled on a 16-bit byte. In fact, the emerging world standard for text (Unicode) uses 16-bit character codes, and allows distinct values for every symbol in every popular writing system.
257 is a Fermat number, a number of the form 22N+1. Fermat hypothesized that all of these were prime, but they are not. A discussion of them and their factors is here. See also 641.
(Personal: the course number of my only Men's Weekend production)
The number of days in the tzol k'in, the almanac cycle Mayans used for divination (a system that originated at least 2600 years ago and is still used by native Americans in Yucatan, Guatemala, Belize, and Honduras). The days in the tzol k'in have successive names from a set of 20 (similar to the 20 days in the Mayan civil week) and numbers from a set of 13. There are thus 13×20 combinations of a name and a number and it takes this many days for the cycle to repeat. It is not known why 260 was "chosen" although there are many theories. It is well-known the Mayans recognized a period of 52×365 days (slightly less than 52 tropical years), which is exactly equal to 18980=73×260 — but this does not line up with other important astronomical periods (the eclipse seasons, the lunar month or the synodic period of Venus). See also 5126 and 1872000.
See 220.
This is the value of "4 superfactorial" by the lower (Simon and Plouffe 1995) definition of "superfactorial".
Barnes' G-function is to superfactorials as the Gamma function is to normal factorials. Barnes' G-function can be (very slowly) calculated by the formula:
G(z) = 2πz/2 e-[z(z+1)+γ z2]/2
PRODUCT(n=1..inf)[(1+z/n)n e-z+z2/(2n)]
where γ is the Euler-Mascheroni constant. For sufficiently large values of z you can use the approximation:
G(n) ≅ (e1/12/A) nn2/2-1/12(2π)n/2e-3n2/4
where A is the Glaisher-Kinkelin constant. See the MathWorld page 86 for more.
See also 1010102.0691973765×1036305. There is also a hyperfactorial.
I recently moved, and my house number changed from 27 to 304. I wanted to find relationships between these two numbers, and quickly found two rather straightforward ones:
304 = 3 × 10 × 10 + 4 and 27 = 3 + 10 + 10 + 4
(change × to +)
304 = 24 × 19 and 27 = 2 × 4 + 19 (change ^ to × and × to +)
This sort of thing is very common.
323 is a Motzkin number; see 51.
323 = 17×19, a product of two primes that, together with 1001 and 2001, create the rather nice digit pattern of the primorial 6469693230.
325 = 182+12 = 172+62 = 152+102, and is the smallest number that can be expressed as the sum of two distinct positive squares in at least 3 different ways. The sequence of such numbers is: 5, 65, 325, 1105, 5525, 27625, 71825, ... (Sloane's A052199; see also A093195). An interesting thing about these numbers is that all of their factors are prime numbers of the form 4n+1 (5, 13, 17, 29, 37, ... the "Pythagorean primes" A002144), and their factorizations are similar to those of the highly composite numbers. See also 1729 and 635318657.
327 is a perfect totient number. This is like a perfect number, but adding together the iterates of the totient function instead of adding together the number's factors. The Euler totient function, Sloane's A000010, counts how many numbers smaller than N are relatively prime to N. For example, if N is 14, the numbers 1, 3, 5, 9, 11 and 13 are relatively prime (notice that 1 counts), so the totient function of 14 is 6. For a prime P we're counting all numbers smaller than P, which is P-1, so the totient function always gives a smaller number except (by definition) when N=1. So, we can take a number like 327, and compute the Totient function (which gives 216), then compute the totient function of that number, and so on (giving 72, 24, 8, 4, 2 and 1), then add all these numbers together and we get 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327. There is more about these numbers on a separate page here; see also 243.
327 also "combines" two of my favorite numbers, 3 and 27. When watching a baseball game I immediately notice when a batter's average or some similar statistic is .327, but it seems to happen more often then random chance would allow (.327 results from a record of 17 for 52, or a multiple like 34 for 104). This is the cult number effect in action.
A "3-dimensional oblong number", the volume of a rectangular solid of dimensions 6×7×8. If you add the middle number (7) you get 73=343. Numbers of this type are always a multiple of 6, because in any three consecutive integers there is always exactly one multiple of 3 and at least one even number. The sequence starts: 6, 24, 60, 120, 210, 336, 504, 720, 990, 1320, 1716, ... (Sloane's A007531).
A "generalized oblong number" is the product of 2 or more consecutive positive integers. These include the regular oblong numbers, 3-way products such as 336, and numbers like 120 formed by the product of 4 or more integers. The sequence starts: 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, ... (Sloane's A045619). Each is the value of nPr for some values of n and r, n>r and r>1.
For all prime numbers p (except 2), 2p-1 modulo p is 1:
3 is prime, 23-1 = 4 = 3×1+1
5 is prime, 25-1 = 16 = 5×3+1
7 is prime, 27-1 = 64 = 7×9+1
11 is prime, 211-1 = 1024 = 11×93+1
13 is prime, 213-1 = 4096 = 13×315+1
etc.
This is also true for some (but only a very few) composite numbers. The first composite number for which it is true is 341 = 11 × 31.
Checking a number p by this method is called a primality test because it allows you to quickly find out if a number is composite. The same method works with other bases in place of 2, as long as the base and p have no common factors. With base 2 you just have to check if p is odd or even. 341 "fools the test" for base 2 because it gives the same result a prime number would: 2340 is 1 modulo 341. However, it doesn't fool the test for base 3: 3340 is 56 modulo 341. See 561 for more.
341 is also 143 backwards.
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