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# 666 (number)

There is also a house music group named 666 . For the year, see AD 666.

666 (Six hundred sixty-six in American English, Six hundred and sixty-six elsewhere) is the Number of the Beast in most versions of the Christian Bible, in the Book of Revelation.

666
CardinalSix hundred and sixty-six
Ordinal666th
Factorization$2 \cdot 3^2 \cdot 37$
Roman numeralDCLXVI
Binary1010011010
Base32KQ
Base36 II

## Mathematics

Like any number, 666 has many mathematical properties, some more interesting than others:

• 666 = 2 × 32 × 37 is the 333rd even number and the 544th composite number. (The 666th composite number is 806 = 2 × 13 × 31). 666 is the 260th non square-free number: {4, 8, 9, 12, 16, 18, 20, 24, ..., 648, 650, 652, 656, 657, 660, 664, 666, ...}. The 666th non square-free number is 1701.
• 666 is a composite number (being quite evidently divisible by 2, 3 and 111). The 666th prime is 4973. 666 is not a lucky number. The 666th lucky number is 5559.
• 666 has 12 divisors: {1, 2, 3, 6, 9, 18, 37, 74, 111, 222, 333, 666}. The sum of its divisors is 1482. The first numbers in the interval [1, 106] that have the same sum of divisors are 666, 1017, 1481. The sum of its proper divisors is 816>666, so 666 is the 162nd abundant number. 666 has 3 distinct prime divisors {2, 3, 37} (ω(666) = ω(2 × 32 &times 37) = ω(2) + ω(32) + ω(37) = 1+1+1=3) and a total of 4 prime divisors (Ω(666)=...=4).
• By Euler's totient function φ(666) = 216. 666 is the 14th number with this value of φ: {247, 259, 327, 333, 351, 399, 405, 436, 494, 518, 532, 648, 654, 666, ...} But there is no number whose value of φ is 666, making it a nontotient.
• 666 is the 76th palindromic number in base 10 and it is also the sum of two consecutive palindromic primes: 666 = 313 + 353. It is the 62nd palindromic composite number.
• 666 is the sum of the squares of the first seven primes.
• The sum of all the integers from 1 to 36 inclusive is 666. This means 666 is the 36th triangular number. (The 666th triangular number, on the other hand, (1 + 2 + 3 + ... + 664 + 665 + 666) is 222111.)
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• Since 36 is both square and triangular, 666 is the sixth number of the form n2(n2 + 1) / 2 (triangular squares) and the eighth number of the form n(n + 1)(n2 + n + 2) / 8 (doubly triangular numbers.)
• $6^{6^6}$ or 646656 is 36,306 digits long. Its first 150 digits are:
26591197721532267796824894043879185949053422002699
24300660432789497073559873882909121342292906175583
03244068282650672342560163577559027938964261261109...
The sum of the digits 1 through 146 of this number is 666, as is the sum of its digits 2 through 148 and its digits 2 through 149.
• It is also curious that the Roman numeral representation of the number 666 (DCLXVI) uses once each of the Roman numeral symbols with values under 1,000, and that they are in exact reverse order of their respective values (D = 500, C = 100, L = 50, X = 10, V = 5, I = 1). Curious indeed that the Roman Empire was the most prominent empire in the world at the time this number originally became symbolically attached to "the Devil".
• On the Windows operating system, the key sequence Alt+666 will create a capital U umlaut (Ü).
• 666 is also the value of the frequency of the Bodenseesender, a big AM radio station in South Germany, which is receivable at night in all of Europe, in Kilohertz.
• In the United States, 666 is a brand name (or should we say brand number?) of cough syrup.
• In UNIX and similar operating systems, file permissions are granted by a triplet of octal numerals, 000 through 777. Each position denotes the group of users for which the permission applies: the first digit for owner, second digit for group, and third digit for all other users. 4 grants read permission; 2 grants write permission; and 1 grants execute permission. Permissions are simply summed for each position. For example, a file permission of 751 would grant the owner of the file the permission to read, write, and execute the file; users in the file owner's group to read and execute the file; and all other users to execute the file only. Thus, a file permission of 666 (which is not uncommon) grants all users read and write permissions on the file.
• 666 is the sum of all the numbers on a typical roulette wheel.
03-10-2013 05:06:04