69 (number)

69 (sixty-nine; LXIX) is the natural number following 68 and preceding 70. An odd number and a composite number, 69 is divisible by 1, 3, 23 and 69. 69 is a semiprime because it is a natural number that is the product of exactly two prime numbers (3 and 23), and an interprime between the numbers of 67 and 71. Because 69 is not divisible by any square number other than 1, it is categorised as a square-free integer. 69 is also a Blum integer since the two factors of 69 are both Gaussian primes. In number theory, 69 is a deficient number, arithmetic number and a congruent number.

 ← 68 69 70 →
Cardinalsixty-nine
Ordinal69th
(sixty-ninth)
Factorization3 × 23
Divisors1, 3, 23, 69
Greek numeralΞΘ´
Roman numeralLXIX
Binary10001012
Ternary21203
Senary1536
Octal1058
Duodecimal5912

The number and its pictograph give its name to the sexual position of the same name. The association of the number with this sex position has resulted in it being associated in meme culture with sex. People knowledgeable of the meme may respond "nice" in response to the appearance of the number, whether intentionally an innuendo or not.

In mathematics

69, spelled sixty-nine and known in Roman numerals as LXIX,[a] is the natural number that follows 68 and precedes 70.[1][2] An odd number, 69 is divisible by 1, 3, 23 and 69.[3][4] 69 is a composite number, meaning that it is not a prime, and a lucky number because it is a natural number that remains after repeatedly removing every nth number in a sequence of natural numbers, starting from 1.[b][6][7] As a natural number that is the product of exactly two prime numbers (a factorisation of 3 x 23), 69 is a semiprime and an interprime between the prime numbers of 67 and 71.[8][9] The aliquot sum—the total sum of all the divisors of a number, excluding the number itself—of 69 is 27 within the aliquot sequence of numbers where each number is the aliquot sum of the previous number (69, 27, 13, 1, 0). 69 is the third composite number in the 13-aliquot tree, following 27 and 35.[10]

Because 69 is not divisible by any square number other than 1, it is categorised as a square-free integer.[11] 69 is a Blum integer since the two factors of 69 are both Gaussian primes, and an Ulam number—an integer that is the sum of two distinct previously occurring Ulam numbers in a sequence.[c][12][14] 69 is a deficient number because the sum of its proper divisors (excluding itself) is less than the number of itself.[15] As an integer for which the arithmetic mean average of its positive divisors is also an integer, 69 is a arithmetic number.[16] 69 is a congruent number—a positive integer that is the area of a right triangle with three rational number sides—and an amenable number.[17][18] 69 can be expressed as the sum of consecutive positive integers in multiple ways, making it a polite number.[19]

In decimal, 69 is the only natural number whose square (4761) and cube (328509) use every digit from 0–9 exactly once.[20][21] It is also the largest number whose factorial is less than a googol. On many handheld scientific and graphing calculators, 69! (1.711224524×1098) is the highest factorial that can be calculated due to memory limitations.[22] In its binary expansion of 1000101,[23] 69 is equal to 105 octal, while 105 is equal to 69 hexadecimal (this same property can be applied to all numbers from 64 to 69).[24][25] In computing, 69 equates to 2120 in ternary (base-3); 153 in senary (base-6); and 59 in duodecimal (base-12).[26][27][28]

Visually, 69 is a strobogrammatic number because it looks the same when viewed both right-side and upside down.[29] 69 is a centered tetrahedral number, a figurate number that represents a pyramid with a triangular base and all other points arranged in layers above the base, forming a tetrahedron shape.[30] 69 is also a pernicious number because there is a prime number of 1s when it is written as a binary number, and an odious number as it is a positive integer that has an odd number of 1s in its binary expansion.[31][32]

In other fields

In chemistry, 69 is the atomic number of thulium, a rare lanthanide (category of metallic elements).[33] In astronomy, the Messier object M69 is a globular cluster in the constellation of Sagittarius;[34] 69 Hesperia is a main-belt asteroid.[35] NGC 69 is the designation given to a barred lenticular galaxy located in the Andromeda constellation.[36][37] In ASCII, 69 is the decimal for the uppercase E character.[25]

69ing is a sex position wherein each partner aligns themselves to simultaneously achieve oral sex with each other.[38] In reference to this sex act, the number 69 itself has become an Internet meme as an inherently funny number in which users will respond to any occurrence of the number with the word "nice" to draw specific attention to it.[39] This means to humorously imply that the reference to the sex position was intentional. Because of its association with the sex position and resulting meme, 69 has been named "the sex number".[39] In music, the American rapper 6ix9ine (pronounced "six nine") chose the stage name in reference to the sex position as well as the yin-yang symbol.[40]

Explanatory footnotes

1. ^ Greek numerals: ΞΘ´
2. ^ Where n is the next number in the list after the last surviving number; every second number (all even numbers) in the list of numbers (1 through infinity) is eliminated first (1, 3, 5, 7, 9, 11 …), every third number (1, 3, 7, 9 …), then every seventh number, and so forth.[5]
3. ^ As a consequence of the definition of the Ulam sequence, 3 is an Ulam number (1 + 2) and 4 is an Ulam number (1 + 3 ). 5 is not an Ulam number, because 5 = 1 + 4 = 2 + 3. 69 is an Ulam number as the sum of 16 + 53; both 16 and 53 are Ulam numbers.[12][13]

References

1. ^ "sixty-nine, n.". Collins English Dictionary. HarperCollins. n.d. Retrieved 22 April 2024.
2. ^ Neil, Sloane; Forgues, Daniel (7 October 2009). "A000027: The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous". On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
3. ^ Neil, Sloane (9 May 2022). "A005408: The odd numbers". On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 23 April 2024.
4. ^ Anjema, Henry (1767). Table of divisors of all the natural numbers from 1. to 10000. p. 2. ISBN 9781140919421 – via the Internet Archive.
5. ^ Giblin, P[eter] J. (1993). Primes and Programming. Cambridge University Press. p. 67. ISBN 9780521409889.
6. ^ Neil, Sloane (16 December 2010). "A002808: Composite numbers". On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
7. ^ Neil, Sloane (7 March 2008). "A000959: Lucky numbers". On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
8. ^ Neil, Sloane; Guy, R. K. (22 August 2010). "A001358: Semiprimes (or biprimes): products of two primes". On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
9. ^ Kimberling, Clark (n.d.). "A024675: Average of two consecutive odd primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
10. ^ Sloane, Neil (n.d.). "A001065: Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n." The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
11. ^ Sloane, Neil (n.d.). "A005117: Squarefree numbers: numbers that are not divisible by a square greater than 1". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
12. ^ a b Gupta, Shyam Sunder (2009). "Smarandache sequence of Ulam numbers". In Wenpeng, Zhang (ed.). Research on Number Theory and Smarandache Notions: Proceedings of the Fifth International Conference on Number Theory and Smarandache Notions. Hexis. p. 78. ISBN 9781599730882.
13. ^ Recaman, Bernardo (1973). "Questions on a sequence of Ulam". American Mathematical Monthly. 80 (8). Mathematical Association of America: 919–920. doi:10.2307/2319404. JSTOR 2319404.
14. ^ Wilson, Robert G. (n.d.). "A016105: Blum integers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
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16. ^ Sloane, Neil; Bernstein, Mira (3 April 2006). "A003601: Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
17. ^ Alter, Ronald; Curtz, Thaddeus B. (January 1974). "A Note on Congruent Numbers". Mathematics of Computation. 28 (125). American Mathematical Society: 304–305. doi:10.2307/2005838. JSTOR 2005838.
18. ^ Beedassy, Lekraj (7 January 2005). "A100832: Amenable numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
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20. ^ Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers (2 ed.). Penguin Books. p. 100. ISBN 0-14-008029-5.
21. ^ Barbeau, Edward (1997). Power Play. Mathematical Association of America. p. 126. ISBN 9780883855232.
22. ^ Brannan, David Alexander (2006). A First Course in Mathematical Analysis. Cambridge University Press. p. 303. ISBN 9781139458955.
23. ^ Konheim, Alan G. (2007). Computer Security and Cryptography. Wiley. p. 382. ISBN 9780470083970.
24. ^ Topham, Douglas W. (2012). A System V Guide to UNIX and XENIX. Springer New York. p. 78. ISBN 9781461232469.
25. ^ a b Holmay, Patrick (1998). "ASCII Character Set (Continued)". The OpenVMS User's Guide. Elsevier Science. p. 272. ISBN 9781555582036.
26. ^ Clifford, Jerrold R.; Clifford, Martin (1974). Computer Mathematics Handbook. Allyn & Bacon. p. 276.
27. ^ Scott, Norman Ross (1960). Analog and Digital Computer Technology. McGraw-Hill. p. 221.
28. ^ Meyer, Jerome S. (1963). More Fun with Mathematics. Gramercy Publishing Company. p. 73.
29. ^ Deza, Elena (2013). Perfect And Amicable Numbers. World Scientific. p. 390. ISBN 9789811259647.
30. ^ Deza, Elena; Deza, Michel (2012). Figurative Numbers. World Scientific. pp. 126–127. ISBN 9789814355483.
31. ^ Gow, Jeremy (8 February 2000). "A052294: Pernicious numbers: numbers with a prime number of 1's in their binary expansion". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
32. ^ Sloane, Neil (n.d.). "A000069: Odious numbers: numbers with an odd number of 1's in their binary expansion". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 April 2024.
33. ^ Stwertka, Albert (2002). A Guide to the Elements (2 ed.). Oxford University Press. p. 161. ISBN 9780195150261.
34. ^ Kitchin, C. R. (2012). Illustrated Dictionary of Practical Astronomy. Springer London. p. 262. ISBN 9781447101758.
35. ^ Shepard, Michael K.; Harris, Alan W.; Taylor, Patrick A.; Clark, Beth Ellen; Ockert-Bell, Maureen; Nolan, Michael C.; et al. (3 August 2011). "Radar observations of Asteroids 64 Angelina and 69 Hesperia" (PDF). Icarus. 215 (2). Elsevier: 547–551. arXiv:1104.4114. Bibcode:2011Icar..215..547S. doi:10.1016/j.icarus.2011.07.027. Retrieved 22 April 2024 – via NASA.
36. ^ "NGC 69". Students for the Exploration and Development of Space. n.d. Retrieved 22 April 2024.
37. ^ Steinicke, Wolfgang (2010). Observing and Cataloguing Nebulae and Star Clusters: From Herschel to Dreyer's New General Catalogue. Cambridge University Press. p. 191. ISBN 9781139490108.
38. ^ Coleman, Julia (2022). Love, Sex, and Marriage: A Historical Thesaurus. Brill Publishers. p. 214. ISBN 9789004488502.
39. ^ a b Feldman, Brian (9 June 2016). "Why 69 Is the Internet's Coolest Number (Sex)". Intelligencer. Retrieved 22 April 2024.
40. ^ Witt, Stephen (16 January 2019). "Tekashi 69: The Rise and Fall of a Hip-Hop Supervillain". Rolling Stone. Retrieved 23 April 2024.