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343 equals 73 = 35 + 102
343 = (3+4)3, a "Friedman number"[96]
and (tragically) it is the number of New York City firefighters killed at the World Trade Center on 2001 Sep 11th.
360 can be evenly factored in 24 different ways — more than any other number of this size (see this entry for more about this).
360 is the number of degrees in a full circle. This size for angular division was chosen by the Babylonians because it is 602/10 and because it is close to the distance the sun moves through the zodiac each day (in other words, it is close to the length of the tropical year. In N days, the sun moves just about N degrees across the zodiac.
Length of the Harvard Bridge in Boston (which actually bridges Boston to M.I.T. and is nowhere near Harvard University), measured in "smoots". In October 1958, as a fraternity prank, then-freshman Oliver Smoot (MIT class of 1962) was used as a unit of measurement to measure the length of the bridge (which MIT students use to walk to and from Boston.) Appropriately, Smoot went on in life to eventually serve as President of the International Organization for Standardization (ISO), and as chairman of the American National Standards Institute (ANSI).
His height at the time, 5 feet 7 inches, is a fairly well known "nonstandard unit" of length measurement. It is available in Google's built-in calculator (type "27 feet in smoots" in any Google search box) and as an optional unit in Google Earth (in the Ruler tool). The markings themselves are so well-accepted by the public that when the bridge was replaced in the 1980s, the Cambridge Police department requested that the markings be maintained, since they had become useful for identifying the location of accidents on the bridge. The construction crew went one better, by actually scoring the concrete surface of the sidewalk on the bridge at 1-smoot intervals instead of something more conventional, like 4 or 6 feet (see here).
Often used in quick and sloppy calculations as the number of days in a year. 365 is 52 weeks plus one day.
The length of the mean tropical year, in mean solar days. The tropical year is the amount of time for the Earth to complete one orbit around the Sun, measured from equinox to equinox or from solstice to solstice. It differs from the sidereal year because of precession and the effects of other planets. This is the most important definition of "year" for most people because it is the one that relates the frequency of day and night to the frequency of the seasons. It is the value that solar and lunisolar calendars try to approximate. The figure given here is for the year 2000; it decreases by 0.00000006162 each year due to the slowing of Earth's rotation.
Conveniently for us Earthlings, it happens to be pretty darn close to 360, a number with lots of factors. This means that, for lots of fairly common divisions (like 1/3 of a year, 1/4 of a year, etc.) it's pretty easy to remember how many days is involved because it's so close to what you'd get if the number were exactly 360. Thus: a quarter-year is just about 90 days; etc.
This is the approximation that the Gregorian calendar uses for the number of days in a year.
The length of the tropical year under the Julian calendar. This approximation was probably first known to the Egyptians, who had stuck to a calendar of 365-day years (with no leap-years) for a staggeringly long period of time, and were therefore able to measure how inaccurate it was through observation over time (see 1508.0833, the Sothic cycle).
The length of a sidereal year in mean solar days. A sidereal year is the amount of time it takes Earth to complete one orbit when viewed from a nonrotating reference frame (which is usually approximated by averaging together the positions of many distant stars). It differs from the tropical year because of precession.
See also 365.2568983.
An approximate value of the sidereal year at the time of Carl Friedrich Gauss' work on orbital mechanics and regression analysis. It persists as part of the definition of the astronomical unit.
A reader wrote to me and asked (roughly) the following:
What are the odds of seeing the digits 3, 6 and 8 (in any order, but contiguous) on an automobile license plate? If I see 1000 plates during my drive to work, it is unusual to see 10 cars that feature one of the combinations 368, 386, 638, 683, 836 or 863?
Of course, the answer depends a lot on how many license plates have digits at all, and whether there are lots of plates with 4 or more digits in them, etc. You have to consider how many cars there are with each different number of digits, and figure the odds for each.
For plates with 3-digit numbers, the answer is easy: 6 out of 1000. In general, the odds of seeing your N special digits in an N-digit number are N!/10N. (This is assuming the special digits are different from each other. If some of them are the same, the odds go down.)
For plates with more than N digits, the odds are a little more complicated to compute. For example, for 3 digits in a 4-digit number, the odds are 114 out of 10000. This is 6 less than 120/10000, which is what you get if you take the 6/1000 odds for the 3-digit case and double it. The reason is that you get:
- 60 cases with the 3 digits at the beginning of the 4-digit number
- 60 cases with the 3 digits at the end
- minus 6 cases that got counted twice because the first three are a match and the last three are also a match. These are the 6 cases that consist entirely of the 3 special digits, and have the first and last digit the same: 3683, 3863, 6386, 6836, 8368 and 8638.
The general formula for the N+1 case is (2×10-1)N!/10N
As you would expect, the formula keeps getting more complicated. For the N+2 case it's ((3×10-2)×10-1)N!/10N.
370 = (037 + 073 + 307 + 370 + 703 + 730)/6, which is the average of all possible permutations of its digits. It and several other multiples of 37 have this property for reasons that are discussed under that number. A larger example with a cool digit pattern is 456790123. Numbers with this property were first pointed out to me by Claudio Meller. Many larger examples are discussed here.
371 = 7×53 = 7 + 11 + ... + 53. Like 39, it is the product of two primes p1 and p2 and the sum of all primes from p1 to p2 inclusive. The next such number is 454539357304421 (see that entry for more).
371 is also a sum of powers of its own digits; see 153 for more.
This is π4 + π5, notable for being very close to e6 (which is 403.428793...).52
The smallest number that requires more than three terms to express as a sum of 3-smooth numbers, as in X = 2a3b + 2c3d + 2e3f + ... (the next record-settters are 18431, 3448733, and 1441896119). This type of representation of a number is referred to as a "double-base number system" (DBNS); see here.
432 = 24 33 which makes it 3-smooth. Like its factors 108 and 216, it occurs occasionally in religious and spiritual contexts, most often multiplied by some power of 10 (see 43200, 432000, 4320000, and 4320000000). If you keep doubling further, you get 864, 1728, and 3456.
An example of an order-4 Kaprekar number: take an n-digit number, raise it to the 4th power, divide the result into 4 groups of n digits each, add together and get the original number. In the specific case of 433, the 4th power is 35152125121; divide this into groups of 3 digits (because 433 has 3 digits) and add: 35+152+125+121 = 433. A table of larger ones is located here. See also 7776.
464 is the sum of row 7 of the following triangle, which is similar to the fibonomial triangle which in turn resembles Pascal's triangle. I refer to the numbers in this triangle as meta-fibonomials:
These numbers are defined similarly to the fibonomials, but involve terms of the form FFn, where Fn represents the nth Fibonacci number. For example, the 4th element in the 7th row (210) is FF6FF5FF4/FF3FF2FF1 = F8F5F3/F2F1F1 = 21×5×3/1×1×1. The general form is FFn...FFn-k+1/FFk...FF1. This formula always gives an integer, for reasons explained in the fibonomial description.
The second number on each row is the sequence of FFn (Sloane's A007570): 1,1,1,2,5,21,233,10946,5702887,.... The third number on each row is FFn×FFn-1, a sequence that starts: 1, 2, 10, 105, 4893, 2550418, 62423801102, ... The 4th number in each row is FFn×FFn+1×FFn+2/2, a sequence that starts 1, 2, 10, 210, 24465, 53558778, ....
The third perfect number, defined as a number that is equal to the sum of its proper factors (factors smaller than itself): 1+2+4+8+16+31+62+124+248=496. The search for perfect numbers was considered an important problem to the mathematicians of the classical Greek era.
As you can see, in this case of 496, all of the factors are either powers of 2 (the 1, 2, 4, 8 and 16) or 31=25-1 times one of those powers of 2. Euclid, working ca. 300 BC, found this pattern and showed that if P is a prime number and if 2P-1 is also prime, then X given by
X = 2P-1 (2P - 1)
= 2N (2N+1 - 1)
is a perfect number. More recently is was shown that all even perfect numbers fit this pattern.
However, not all numbers that fit the pattern are perfect; see 2047 and 8384512. It is still not known if there are any odd perfect numbers.
Prime numbers of the form 2P-1 are called Mersenne primes. See here for a complete list of known perfect numbers.
See also 6, 28, 8589869056, and 4.4823309×1014471464.
The first Carmichael number. Carmichael numbers are odd composite numbers which cannot be found to be composite using Fermat's Little Theorem, which states that for all prime numbers p, if n is relatively prime to p, then np-1-1 is divisible by p (sometimes stated: np - n is divisible by p, which is an equivalent statement). For a Carmichael number, no matter what n you pick this test will show that it might be prime. See also 1729.
Common letter-number value of the word Torah in Hebrew. There are several Hebrew alphabetic number assignments used for Gematria (numerology) but only one system for the common purpose of communicating a number in ordinary text. It dates back to before they used separate symbols for the digits the way they do now. The assignment is as follows15:
א=1 ב=2 ג=3 ד=4 ה=5 ו=6 ז=7 ח=8 ט=9
י=10 כ=20 ל=30 מ=40 נ=50 ס=60 ע=70 פ=80 צ=90
ק=100 ר=200 ש=300 ת=400
Using that assignment, the value of תורה (Torah) is 611 (400+6+200+5)
Appears instead of 666 in some early Greek manuscripts of the relevant passage of the Bible. See the 666 entry for more.
The first factor of a non-prime Fermat number. Fermat conjectured that the Fermat numbers were all prime, and could not factor 232+1=4294967297. In fact 4294967297 is composite, equal to 641 × 6700417. Euler showed that all factors of a Fermat number 22n+1 must be of the form K2n+1+1. In this case, n is 5 and the factor 641 is equal to 10×25+1+1. For more about this see the Wiki page on Fermat primes.
648 is the smallest number that can be expressed as a ba in two different ways: 3×63 = 2×182. Such numbers are not particularly common: 648, 2048, 4608, 5184, 41472, 52488, 472392, 500000, 524288, 2654208, 3125000, 4718592, ...
648 = 3×63 = 2×182
2048 = 8×28 = 2×322
4608 = 9×29 = 2×482
5184 = 4×64 = 3×123
41472 = 3×243 = 2×1442
52488 = 8×38 = 2×1622
472392 = 3×543 = 2×4862
500000 = 5×105 = 2×5002
524288 = 8×48 = 2×5122
2654208 = 3×963 = 2×11522
3125000 = 8×58 = 2×12502
4718592 = 18×218 = 2×15362
10125000 = 3×1503 = 2×22502
13436928 = 8×68 = 2×25922
21233664 = 4×484 = 3×1923
30233088 = 3×2163 = 2×38882
46118408 = 8×78 = 2×48022
76236552 = 3×2943 = 2×61742
134217728 = 8×88 = 2×81922
169869312 = 3×3843 = 2×92162
344373768 = 8×98 = 3×4863 = 2×131222
402653184 = 24×224 = 3×5123
512000000 = 5×405 = 2×160002
648000000 = 3×6003 = 2×180002
737894528 = 7×147 = 2×192082
800000000 = 8×108 = 2×200002
838860800 = 25×225 = 2×204802
922640625 = 5×455 = 3×6753
1147971528 = 3×7263 = 2×239582
1207959552 = 9×89 = 2×245762
1714871048 = 8×118 = 2×292822
1934917632 = 3×8643 = 2×311042
2754990144 = 4×1624 = 3×9723
3127772232 = 3×10143 = 2×395462
3439853568 = 8×128 = 2×414722
4879139328 = 3×11763 = 2×493922
6525845768 = 8×138 = 2×571222
6973568802 = 18×318 = 2×590492
7381125000 = 3×13503 = 2×607502
See also 344373768.
Main article: The number 666
Here are a few of the purely mathematical properties of 666:
666 is the sum of the squares of the first 7 primes: 22 + 32 + 52 + 72 + 112 + 132 + 172.
666 = 13 + 23 + 33 + 43 + 53 + 63 + 53 + 43 + 33 + 23 + 13. Such sums could be called "hyper-octahedral" numbers, based on a 4-dimensional polyhedron analagous to the octahedron.
It is the 36th triangular number: 1+2+3+...+36 = 666, which seems more significant because 36=6×6. 666 is the largest triangular number that is a repdigit.
See also 10(6.65565×10668)
This is 262, the first square that is a palindrome but whose square root is not a palindrome.
In the English language there are 26 letters, so 676 is the number of combinations of 2 letters when order is distinct. If the standard license plates in your area had 2 letters followed by 4 numbers, there would be 262×104 = 676×10000 = 6760000 possible plates.
According to my classical sequence generator, 695 is the next number after my childhood "favorite" numbers 7, 27 and 127. The formula it finds is: A0 = 0; A1 = -1; AN+1 = N AN - AN + 2AN-1 + N (sequence MCS27694341). This serves as an example of how easy it is to find a sequence formula to match an arbitrary set of numbers. See also 715 and 1011.
The first 7 prime numbers (2, 3, 5, 7, 11, 13, 17) can be arranged to form the factors of 714 (2×3×7×17) and 715 (5×11×13). Because of this, the primorial 2×3×5×7×11×13×17=510510 is also an oblong number 714×715=510510.
In base 714 there are easy tests for divisibility by 9 different primes (the 7 listed above plus 23 and 31 because 23×31=713). See the 14, 21, 29 and 66 entries for more about these properties.
714 and 715 also have the property that their prime factors add up to the same total: 2 + 3 + 7 + 17 = 5 + 11 + 13. Another (smaller) example is 77 and 78: 77=7×11, 78=2×3×13, and 7+11=2+3+13. When one of the pair is divisible by a square, it matters how you count the factors. For example, 24 and 25 are a pair if you count each prime only once: 24=2×2×2×3, 25=5×5, and 2+3=5. 15 and 16 qualify if you count the multiples: 15=3×5, 16=2×2×2×2, and 3+5=2+2+2+2. But the pair 714 and 715 qualify regardless of which way you define it.
Such pairs are called Ruth-Aaron pairs because of Babe Ruth's famous record of 714 career home runs, which was broken in 1974 when Hank Aaron hit his 715th. (Aaron reached 755 before retiring, but the number 715 is almost as commonly associated with him; in 2007 his record was surpassed by Bonds).
Part of a Ruth-Aaron pair, see 714.
Follows 3, 7, 27 and 127 in sequence MCS55651588, which is an alternative solution to the "problem" descibed in the entry for 695 found via my classical sequence generator. Another alternative solution is A136580, which would give 747. (As with 695, these numbers are interesting just by being an example of the ease of finding formulas that match a mystery integer sequence. The choice of this example is just as arbitrary as for 695; see its description. For me, 3 was not a favorite number in childhood, except through its connection to 27.)
Like 120 and 210, 720 can be expressed as a product of consecutive integers in two distinct ways: 2×3×4×5×6 = 8×9×10. You can also add a 7 to both sides and get a similar equation whose value is 5040. Here are the smallest numbers with this property: 120, 210, 720, 5040, 175560, 17297280, 19958400, 259459200, 20274183401472000, 25852016738884976640000, 368406749739154248105984000000, ... The sequence is Sloane's A064224. The sequence is infinite — see 19958400 for details as to why. I also have a page dedicated to these numbers.
One of the better-known numbers that is also a product or brand name: a well-known Boeing aircraft. 747 is also the sum of the first few even factorials: 0!+2!+4!+6! = 1+2+24+720 = 747. See also A136580 and 715.
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