1000 (number)

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"Thousand" redirects here. For the Marvel Comics character, see Thousand (comics). For the Moby song, see Thousand (song).
← 999 1000 1001 →
Cardinalone thousand
Ordinal1000th
(one thousandth)
Factorization23× 53
Divisors1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
Greek numeral,Α´
Roman numeralM
Greek prefixchilia
Latin prefixmilli
Binary11111010002
Ternary11010013
Quaternary332204
Quinary130005
Senary43446
Octal17508
Duodecimal6B412
Hexadecimal3E816
Vigesimal2A020
Base 36RS36
Urduہزار
Tamil௲

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it is often written with a comma separating the thousands unit: 1,000. In analogy to the term century for '100 years' the time lapse of 1,000 years is sometimes termed, after the Greek root, chiliad. A chiliad of other objects means 1,000 of them.

It may also be described as the short thousand in historical discussion of medieval contexts where it might be confused with the Germanic concept of the "long thousand" (1200).

In mathematics[edit]

  • The decimal representation for one thousand is
  • The SI prefix for a thousand is kilo-, with the official symbol k—for instance, prefixed to "metre" or its symbol "m", kilometre or km signifies a thousand metres. As such, people occasionally represent the number in a non-standard notation by replacing the last three zeros of the general numeral with "K": for instance, 30K for 30,000.
  • By the SI writing style, a space can be used as a thousands separator, i.e., to separate the digits of a number at every power of 1 000.
  • The sum of Euler's totient function over the first 57 integers is 1000.
  • Prime Curios! mentions that 1000 is the smallest number that generates three primes in the fastest way possible by concatenation of decremented numbers (1000999, 1000999998997, and 1000999998997996995994993 are prime). The criterion excludes counting the number itself.
  • 1000 is a Harshad number in base 10.

In time[edit]

In popular culture[edit]

  • A grand is a slang term for one thousand units of a given currency, usually dollars or pounds. Several grand can be shortened to Gs.
  • The symbol K is sometimes used for a thousand; for example, in referring to units of salary or in reference to the Y2K computer bug.
  • Especially in the United States, the gambling community often refers to denominations of $1000 as dimes.
  • The idiom "a picture is worth a thousand words".
  • According to an ancient Japanese legend, anyone who folds a thousand origami cranes will be granted a wish by a crane.
  • The thousandth of something is often celebrated, as with other round numbers. A good example is a millennium.

Miscellaneous[edit]

Music[edit]

Selected numbers in the range 1001–1999[edit]

1001 to 1099[edit]

1001 – sphenic number (7 × 11 × 13), pentagonal number, pentatope number
1002 – sphenic number, Mertens function zero, abundant number
1004 – heptanacci number[1]
1005 – Mertens function zero
1008 – divisible by the number of primes below it
1009 – smallest four-digit prime, palindromic in bases 11, 15, 19, 24 and 28: (83811, 47415, 2F219, 1I124, 18128)
1010 – Mertens function zero
1011 – The largest number that 2n contains 101 and doesn't contain 11011, and it is a Harshad number in bases 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75 (and 202 other bases).
1013 – Sophie Germain prime,[2] centered square number,[3] Mertens function zero
1014 – Mertens function zero
1015 – square pyramidal number[4]
1016 – member of the Mian–Chowla sequence,[5] stella octangula number
1018 – Mertens function zero
1019 – Sophie Germain prime,[2] safe prime[6]
1020 – polydivisible number
1022 – Friedman number
1023 – the highest number one can count to on one's fingers using binary; also the magic number used in Global Positioning System signals
1024 – 210, the number of bytes in a kilobyte (in 1999, the IEC coined kibibyte to use for 1024 with kilobyte being 1000, but this convention has not been widely adopted)
1027 – sum of the squares of the first eight primes; can be written from base 2 to base 18 using only the digits 0 to 9.
1028 – sum of totient function for first 58 integers; can be written from base 2 to base 18 using only the digits 0 to 9.
1029 – can be written from base 2 to base 18 using only the digits 0 to 9.
1031 – Sophie Germain prime,[2] super-prime
1033 – locale ID of English (United States) in (some version of) Windows.[7]
1035 – triangular number,[8] hexagonal number[9]
1049 – Sophie Germain prime,[2] highly cototient number[10]
1051 – centered pentagonal number,[11] centered decagonal number
1056 – pronic number[12]
1060 – sum of the first 25 primes
1063 – super-prime, sum of seven consecutive primes (137 + 139 + 149 + 151 + 157 + 163 + 167)
1071 – heptagonal number[13]
1072 – centered heptagonal number[14]
1079 – every positive integer is the sum of at most 1079 tenth powers.
1080 – pentagonal number[15]
1081 – triangular number,[8] member of Padovan sequence[16]
1086 – Smith number,[17] sum of totient function for first 59 integers
1087 – super-prime, cousin prime, lucky prime,[18] Kynea number[19]
1089 – 332, nonagonal number, centered octagonal number, first natural integer which digits in its decimal expression get reversed when multiplied by 9.[20]
1091 – cousin prime and twin prime
1092 – divisible by the number of primes below it
1093 – the smallest Wieferich prime (the only other known Wieferich prime is 3511[21]), twin prime and star number[22]

1100 to 1199[edit]

1102 – sum of totient function for first 60 integers
1103 – Sophie Germain prime,[2] balanced prime[23]
1104 – Keith number[24]
1105 – Carmichael number,[25] magic constant of n × n normal magic square and n-queens problem for n = 13, decagonal number,[26] centered square number,[3] 1105 = 332 + 42 = 322 + 92 = 312 + 122 = 232 + 242
1116 – divisible by the number of primes below it
1122 – pronic number,[12] divisible by the number of primes below it
1123 – balanced prime[23]
1124 – Leyland number[27]
1128 – triangular number,[8] hexagonal number,[9] divisible by the number of primes below it
1134 - divisible by the number of primes below it
1138 – recurring number in the works of George Lucas and his companies, beginning with his first feature film – THX 1138; particularly, a special code for Easter eggs on Star Wars DVDs.
1140 – tetrahedral number[28]
1151 – first prime following a prime gap of 22.[29]
1152 – highly totient number[30]
1153 – super-prime, Proth prime[31]
1156 – 342, octahedral number,[32] centered pentagonal number,[11] centered hendecagonal number.[33]
1159 – member of the Mian–Chowla sequence[5]
1161 – sum of the first 26 primes
1162 – pentagonal number,[15] sum of totient function for first 61 integers
1169 – highly cototient number[10]
1170 – highest possible score in a National Academic Quiz Tournaments (NAQT) match
1171 – super-prime
1176 – triangular number[8]
1177 – heptagonal number[13]
1184 – amicable number with 1210[34]
1187 – safe prime,[6] Stern prime,[35] balanced prime[23]
1190 – pronic number[12]
1192 – sum of totient function for first 62 integers
1198 – centered heptagonal number[14]

1200 to 1299[edit]

1200 – the long thousand, ten "long hundreds" of 120 each, the traditional reckoning of large numbers in Germanic languages, the number of households the Nielsen ratings sample[36]
1201 – centered square number,[3] super-prime, centered decagonal number
1210 – amicable number with 1184[37]
1213 – emirp
1216 – nonagonal number[38]
1217 – super-prime, Proth prime[31]
1219 – Mertens function zero
1220 – Mertens function zero
1223 – Sophie Germain prime,[2] balanced prime, 200th prime number[23]
1225 – 352, triangular number, square triangular number,[39] hexagonal number,[9] centered octagonal number[40]
1228 – sum of totient function for first 63 integers
1229 – Sophie Germain prime,[2] number of primes between 0 and 10000
1233 – 122 + 332
1237 – prime of the form 2p-1
1240 – square pyramidal number[4]
1241 – centered cube number[41]
1242 – decagonal number[26]
1247 – pentagonal number[15]
1249 – emirp, trimorphic number[42]
1255 – Mertens function zero
1256 – Mertens function zero
1258 – Mertens function zero
1259 – highly cototient number[10]
1260 – highly composite number,[43] pronic number,[12] the smallest vampire number,[44] sum of totient function for first 64 integers, this number appears twice in the Book of Revelation
1261 – star number,[22] Mertens function zero
1264 – sum of the first 27 primes
1266 – centered pentagonal number,[11] Mertens function zero
1270 – Mertens function zero
1275 – triangular number,[8] sum of the first 50 natural numbers
1279 – Mertens function zero
1280 – Mertens function zero
1282 – Mertens function zero
1283 – safe prime[6]
1285 – Mertens function zero
1288 – heptagonal number[13]
1289 – Sophie Germain prime,[2] Mertens function zero
1291 – Mertens function zero
1292 – Mertens function zero
1296 – 64, 362, sum of the cubes of the first eight positive integers, the number of rectangles on a normal 8 × 8 chessboard, also the maximum font size allowed in Adobe InDesign
1297 – super-prime, Mertens function zero
1299 – Mertens function zero

1300 to 1399[edit]

1300 – Sum of the first 4 fifth powers, mertens function zero, largest possible win margin in an NAQT match
1301 – centered square number[3]
1302 – Mertens function zero
1306 – Mertens function zero. In base 10, raising the digits of 1306 to powers of successive integers equals itself: 1306 = 11 + 32 + 03 + 64. 135, 175, 518, and 598 also have this property.
1307 – safe prime[6]
1308 – sum of totient function for first 65 integers
1309 – the first sphenic number followed by two consecutive such number
1312 – member of the Mian–Chowla sequence;[5] code for "ACAB" itself an acronym for "all cops are bastards"[45]
1318 – Mertens function zero
1319 – safe prime[6]
1325 – Markov number[46]
1326 – triangular number,[8] hexagonal number,[9] Mertens function zero
1327 – first prime followed by 33 consecutive composite numbers
1328 – sum of totient function for first 66 integers
1329 – Mertens function zero
1330 – tetrahedral number,[27] forms a Ruth–Aaron pair with 1331 under second definition
1331 – 113, centered heptagonal number,[14] forms a Ruth–Aaron pair with 1330 under second definition. This is the only cube of the form x2 + x − 1, for x = 36.
1332 – pronic number[12]
1335 – pentagonal number,[15] Mertens function zero
1336 – Mertens function zero
1337 – Used in the novel form of spelling called leet. Approximate melting point of gold in kelvins.
1338 – Mertens function zero
1342 – Mertens function zero
1350 – nonagonal number[38]
1361 – first prime following a prime gap of 34,[29] centered decagonal number
1365 – pentatope number[47]
1367 – safe prime,[6] balanced prime, sum of three, nine, and eleven consecutive primes (449 + 457 + 461, 131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 + 173, and 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151),[23]
1369 – 372, centered octagonal number[40]
1371 – sum of the first 28 primes
1378 – triangular number[8]
1379 – magic constant of n × n normal magic square and n-queens problem for n = 14.
1381 – centered pentagonal number[11]
1387 – 5th Fermat pseudoprime of base 2,[48] 22nd centered hexagonal number and the 19th decagonal number,[26] second Super-Poulet number.[49]
1394 – sum of totient function for first 67 integers
1395 – vampire number,[44] member of the Mian–Chowla sequence[5]

1400 to 1499[edit]

1404 – heptagonal number[13]
1405 – 262 + 272, 72 + 82 + … + 162, centered square number[3]
1406 – pronic number,[12] semi-meandric number[50]
1409 – super-prime, Sophie Germain prime,[2] smallest number whose eighth power is the sum of 8 eighth powers, Proth prime[31]
1419 – Zeisel number[51]
1425 – self-descriptive number in base 5
1426 – sum of totient function for first 68 integers, pentagonal number[15]
1430 – Catalan number[52]
1431 – triangular number,[8] hexagonal number[9]
1432 – member of Padovan sequence[16]
1433 – super-prime, Typical port used for remote connections to Microsoft SQL Server databases
1435 – vampire number;[44] the standard railway gauge in millimetres, equivalent to 4' 8½"
1439 – Sophie Germain prime,[2] safe prime[6]
1440 – a highly totient number[30] and a 481-gonal number. Also, the number of minutes in one day, the blocksize of a standard 3 1/2 floppy disk, and the horizontal resolution of WXGA(II) computer displays
1441 – star number[22]
1444 – 382, smallest pandigital number in Roman numerals
1447 – super-prime, happy number
1451 – Sophie Germain prime[2]
1458 – maximum determinant of an 11 by 11 matrix of zeroes
1459 – Sexy prime with 1453, sum of nine consecutive primes (139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181), pierpont prime
1469 – octahedral number,[32] highly cototient number[10]
1470 – pentagonal pyramidal number,[53] sum of totient function for first 69 integers
1471 – super-prime, centered heptagonal number[14]
1480 – sum of the first 29 primes
1481 – Sophie Germain prime[2]
1482 – pronic number[12]
1485 – triangular number
1487 – safe prime[6]
1490 – tetranacci number[54]
1491 – nonagonal number,[38] Mertens function zero
1492 – Mertens function zero
1493 – Stern prime[35]
1494 – sum of totient function for first 70 integers
1496 – square pyramidal number[4]
1499 – Sophie Germain prime,[2] super-prime

1500 to 1599[edit]

1501 – centered pentagonal number[11]
1510 – deficient number, odious number
1511 – Sophie Germain prime,[2] balanced prime[23]
1513 – centered square number[3]
1518 – Mertens function zero
1519 – Mertens function zero
1520 – pentagonal number,[15] Mertens function zero, forms a Ruth–Aaron pair with 1521 under second definition
1521 – 392, Mertens function zero, centered octagonal number,[40] forms a Ruth–Aaron pair with 1520 under second definition
1523 – super-prime, Mertens function zero, safe prime,[6] member of the Mian–Chowla sequence[5]
1524 – Mertens function zero
1525 – heptagonal number,[13] Mertens function zero
1527 – Mertens function zero
1528 – Mertens function zero
1530 – vampire number[44]
1531 – centered decagonal number, Mertens function zero
1532 – Mertens function zero
1535 – Thabit number
1536 - a common size of microplate
1537 – Keith number,[24] Mertens function zero
1540 – triangular number, hexagonal number,[9] decagonal number,[26] tetrahedral number[27]
1543 – Mertens function zero
1544 – Mertens function zero
1546 – Mertens function zero
1556 – sum of the squares of the first nine primes
1559 – Sophie Germain prime[2]
1560 – pronic number[12]
1564 – sum of totient function for first 71 integers
1572 – member of the Mian–Chowla sequence[5]
1575 – odd abundant number[55]
1583 – Sophie Germain prime
1588 – sum of totient function for first 72 integers
1593 – sum of the first 30 primes
1596 – triangular number
1597 – Fibonacci number,[56] Markov number,[46] super-prime, emirp

1600 to 1699[edit]

1600 – 402, repdigit in base 7 (44447), street number on Pennsylvania Avenue of the White House, length in meters of a common High School Track Event, perfect score on SAT (except from 2005-2015)
1601 – Sophie Germain prime, Proth prime,[31] the novel 1601 (Mark Twain)
1617 – pentagonal number[15]
1618 – centered heptagonal number[14]
1619 – safe prime[6]
1621 – super-prime
1625 – centered square number[3]
1626 – centered pentagonal number[11]
1633 – star number[22]
1638 – harmonic divisor number[57]
1639 – nonagonal number[38]
1640 – pronic number[12]
1649 – highly cototient number,[10] Leyland number[27]
1651 – heptagonal number[13]
1653 – triangular number, hexagonal number[9]
1657 – cuban prime,[58] prime of the form 2p-1
1660 – sum of totient function for first 73 integers
1666 – largest efficient pandigital number in Roman numerals (each symbol occurs exactly once)
1669 – super-prime
1679 – highly cototient number,[10] semiprime (23 × 73, see also Arecibo message)
1680 – highly composite number[43]
1681 – 412, smallest number yielded by the formula n2 + n + 41 that is not a prime; centered octagonal number[40]
1682 – member of a Ruth–Aaron pair (first definition)
1683 – member of a Ruth–Aaron pair (first definition)
1695 – magic constant of n × n normal magic square and n-queens problem for n = 15.
1696 – sum of totient function for first 74 integers

1700 to 1799[edit]

1701 – decagonal number, hull number of the U.S.S. Enterprise on Star Trek
1702 – palindromic in 3 consecutive bases: 89814, 78715, 6A616
1705 – tribonacci number[59]
1709 – first of a sequence of eight primes formed by adding 57 in the middle. 1709, 175709, 17575709, 1757575709, 175757575709, 17575757575709, 1757575757575709 and 175757575757575709 are all prime, but 17575757575757575709 = 232433 × 75616446785773
1711 – triangular number, centered decagonal number
1717 – pentagonal number[15]
1720 – sum of the first 31 primes
1722 – Giuga number,[60] pronic number[12]
1723 – super-prime
1728 – the quantity expressed as 1000 in duodecimal, that is, the cube of twelve (called a great gross), and so, the number of cubic inches in a cubic foot, palindromic in base 11 (133111) and 23 (36323)
1729 – taxicab number, Carmichael number, Zeisel number, centered cube number, Hardy–Ramanujan number. In the decimal expansion of e the first time all 10 digits appear in sequence starts at the 1729th decimal place. In 1979 the rock musical Hair closed on Broadway in New York City after 1729 performances. Palindromic in bases 12, 32, 36.
1733 – Sophie Germain prime, palindromic in bases 3, 18, 19.
1736 – sum of totient function for first 75 integers
1741 – super-prime, centered square number[3]
1747 – balanced prime[23]
1753 – balanced prime[23]
1756 – centered pentagonal number[11]
1760 – the number of yards in a mile
1764 – 422
1770 – triangular number, hexagonal number,[9] Town of Seventeen Seventy in Australia
1771 – tetrahedral number[27]
1772 – centered heptagonal number,[14] sum of totient function for first 76 integers
1782 – heptagonal number[13]
1785 – square pyramidal number[4]
1787 – super-prime, sum of eleven consecutive primes (137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 + 191)
1791 – largest natural number that cannot be expressed as a sum of at most four hexagonal numbers.
1794 – nonagonal number[38]

1800 to 1899[edit]

1800 – pentagonal pyramidal number,[53] Achilles number, also, in da Ponte's Don Giovanni, the number of women Don Giovanni had slept with so far when confronted by Donna Elvira, according to Leporello's tally
1801 – cuban prime, sum of five and nine consecutive primes (349 + 353 + 359 + 367 + 373 and 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227)[58]
1806 – pronic number,[12] product of first four terms of Sylvester's sequence, primary pseudoperfect number,[61] only number for which n equals the denominator of the nth Bernoulli number[62]
1807 – fifth term of Sylvester’s sequence[63]
1811 – Sophie Germain prime
1820 – pentagonal number,[15] pentatope number[47]
1821 – member of the Mian–Chowla sequence[5]
1823 – super-prime, safe prime[6]
1827 – vampire number[44]
1828 – meandric number, open meandric number
1830 – triangular number
1832 – sum of totient function for first 77 integers
1834 – octahedral number,[32] sum of the cubes of the first five primes
1836 – factor by which a proton is more massive than an electron
1837 – star number[22]
1841 – Mertens function zero
1843 – Mertens function zero
1844 – Mertens function zero
1845 – Mertens function zero
1847 – super-prime
1849 – 432, palindromic in base 6 (= 123216), centered octagonal number[40]
1851 – sum of the first 32 primes
1853 – Mertens function zero
1854 – Mertens function zero
1856 – sum of totient function for first 78 integers
1857 – Mertens function zero
1861 – centered square number,[3] Mertens function zero
1862 – Mertens function zero, forms a Ruth–Aaron pair with 1863 under second definition
1863 – Mertens function zero, forms a Ruth–Aaron pair with 1862 under second definition
1864 – Mertens function zero
1866 – Mertens function zero
1870 – decagonal number[26]
1885 – Zeisel number[51]
1889 – Sophie Germain prime, highly cototient number[10]
1891 – triangular number, hexagonal number,[9] centered pentagonal number[11]
1892 – pronic number[12]
1896 – member of the Mian–Chowla sequence[5]
1897 – member of Padovan sequence[16]

1900 to 1999[edit]

1900 – 1900 (film) or Novecento, 1976 movie
1901 – Sophie Germain prime, centered decagonal number
1907 – safe prime,[6] balanced prime[23]
1909 – hyperperfect number[64]
1913 – super-prime
1918 – heptagonal number[13]
1926 – pentagonal number[15]
1929 – Mertens function zero
1931 – Sophie Germain prime
1933 – centered heptagonal number,[14] prime number
1934 – sum of totient function for first 79 integers
1936 – 442, 18-gonal number,[65] 324-gonal number.
1938 – Mertens function zero
1951 – cuban prime[58]
1953 – triangular number
1956 – nonagonal number[38]
1966 – sum of totient function for first 80 integers
1969 – Only value less than four million for which a "mod-ification" of the standard Ackermann Function does not stabilize[66]
1973 – Sophie Germain prime
1980 – pronic number[12]
1984 – 11111000000 in binary, see also: 1984 (disambiguation)
1985 – centered square number[3]
1987 – 300th prime number
1988 – sum of the first 33 primes

References[edit]

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  7. ^ [1].
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  21. ^ Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 163
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  30. ^ a b "Sloane's A097942 : Highly totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  31. ^ a b c d "Sloane's A080076 : Proth primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  32. ^ a b c "Sloane's A005900 : Octahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  33. ^ "Sloane's A069125 : a(n) = (11*n^2 - 11*n + 2)/2". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  34. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 61. ISBN 978-1-84800-000-1.
  35. ^ a b "Sloane's A042978 : Stern primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  36. ^ Meehan, Eileen R., Why TV is not our fault: television programming, viewers, and who's really in control Lanham, MD: Rowman & Littlefield, 2005
  37. ^ Higgins, ibid.
  38. ^ a b c d e f "Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  39. ^ "Sloane's A001110 : Square triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  40. ^ a b c d e "Sloane's A016754 : Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  41. ^ "Sloane's A005898 : Centered cube numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  42. ^ "Sloane's A033819 : Trimorphic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  43. ^ a b "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  44. ^ a b c d e "Sloane's A014575 : Vampire numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  45. ^ "Constitutional Court allows 'FCK CPS' sticker". 28 April 2015. "...state court in Karlsruhe ruled that a banner ... that read 'ACAB' - an abbreviation of 'all cops are bastards' ... a punishable insult. ... A court in Frankfurt ... the numbers '1312' constituted an insult ... the numerals stand for the letters ACAB's position in the alphabet.
  46. ^ a b "Sloane's A002559 : Markoff (or Markov) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  47. ^ a b "Sloane's A000332 : Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  48. ^ "Sloane's A001567 : Fermat pseudoprimes to base 2". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  49. ^ "Sloane's A050217 : Super-Poulet numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  50. ^ "Sloane's A000682 : Semimeanders". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  51. ^ a b "Sloane's A051015 : Zeisel numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  52. ^ "Sloane's A000108 : Catalan numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  53. ^ a b "Sloane's A002411 : Pentagonal pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  54. ^ "Sloane's A000078 : Tetranacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  55. ^ "Sloane's A005231 : Odd abundant numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  56. ^ "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  57. ^ "Sloane's A001599 : Harmonic or Ore numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  58. ^ a b c "Sloane's A002407 : Cuban primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  59. ^ "Sloane's A000073 : Tribonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  60. ^ "Sloane's A007850 : Giuga numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  61. ^ "Sloane's A054377 : Primary pseudoperfect numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  62. ^ Kellner, Bernard C.; ‘The equation denom(Bn) = n has only one solution’
  63. ^ "Sloane's A000058 : Sylvester's sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  64. ^ "Sloane's A034897 : Hyperperfect numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  65. ^ "Sloane's A051870 : 18-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-12.
  66. ^ Jon Froemke & Jerrold W. Grossman (Feb 1993). "A Mod-n Ackermann Function, or What's So Special About 1969?". The American Mathematical Monthly. Mathematical Association of America. 100 (2): 180–183. JSTOR 2323780.
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