Английская Википедия:45 (number)
Шаблон:Infobox number 45 (forty-five) is the natural number following 44 and preceding 46.
In mathematics
Forty-five is the smallest odd number that has more divisors than <math>n+1</math>, and that has a larger sum of divisors than <math>n+1</math>.[1][2] It is the sixth positive integer with a square-prime prime factorization of the form <math>p^{2}q</math>, with <math>p</math> and <math>q</math> prime, and first of the form <math>3^{2} \times q</math>. 45 has an aliquot sum of 33 that is part of an aliquot sequence composed of five composite numbers (45, 33, 15, 9, 4, 3, 1, and 0), all of-which are rooted in the 3-aliquot tree. This is the longest aliquot sequence for an odd number up to 45.
Forty-five is the sum of all single-digit decimal digits: <math>0+1+2+3+4+5+6+7+8+9=45</math>. It is, equivalently, the ninth triangle number.[3]
Forty-five is also the fourth hexagonal number and the second hexadecagonal number, or 16-gonal number.[4][5] It is also the second smallest triangle number (after 1 and 10) that can be written as the sum of two squares.
Forty-five is the smallest positive number that can be expressed as the difference of two nonzero squares in more than two ways: <math>7^2-2^2</math>, <math>9^2-6^2</math> or <math>23^2-22^2</math> (see image).[6]
Since the greatest prime factor of <math>45^{2}+1=2026</math> is 1,013, which is much more than 45 twice, 45 is a Størmer number.[7] In decimal, 45 is a Kaprekar number and a Harshad number.[8][9]
Forty-five is a little Schroeder number; the next such number is 197, which is the 45th prime number.[10]
Forty-five is conjectured from Ramsey number <math>R(5, 5)</math>.[11][12]
<math>\phi(45)=\phi(\sigma(45))</math>[13]
Abstract algebra
In the classification of finite simple groups, the Tits group <math>\mathbb {T}</math> is sometimes defined as a nonstrict group of Lie type or sporadic group, which yields a total of 45 classes of finite simple groups: two stem from cyclic and alternating groups, sixteen are families of groups of Lie type, twenty-six are strictly sporadic, and one is the exceptional case of <math>\mathbb {T}</math>. Inside the largest sporadic group, the Friendly Giant <math>\mathbb {F_{1}}</math>, there exist at least 45 conjugacy classes of maximal subgroups, which includes the double cover of the second largest sporadic group <math>\mathbb {B}</math>.[14]
In science
- The atomic number of rhodium
Astronomy
- Messier object M45, a magnitude 1.4 open cluster in the constellation Taurus, also known as the Pleiades
- The New General Catalogue object NGC 45, a magnitude 10.6 spiral galaxy in the constellation Cetus
In music
- A type of gramophone record classified by its rotational speed of 45 revolutions per minute (rpm)
- The group Stars on 45 and its self-titled 1981 song, "Stars on 45"
- "45 and Fat", a 1996 song by Babybird
- "Forty-Five", the title of a 2000 song by The Atomic Bitchwax
- "45", the title of a 2002 song by Elvis Costello, both referring to the 45 rpm singles and to the artist's age when he wrote the song.
- "45", the title of a 2012 song by The Gaslight Anthem
- "45", the title of a 2006 song by noodles
- "45", the title of a 2007 song by The Saturday Knights
- "45", the title of a 2003 song by Shinedown
- 45, the title of a 1982 album by Kino
- "Do the 45", the title of a 2007 song by Ryan Shaw
- Шаблон:Em is repeated continuously in the lyrics of the 1997 song "Brimful of Asha" by Cornershop
In other fields
Шаблон:Seealso Forty-five may also refer to:
- The '45 refers to the Jacobite rising of 1745 in Great Britain, or the year that World War II ended, which was 1945.
- A card game: Forty-five.
- .45, a 2006 motion picture.
- +45 is the telephone dialing code for Denmark.
- 45, a book of essays by record producer Bill Drummond, derived both from the speed of a pop single and from his age when he finished writing it.
- A football match consists of two periods of 45 minutes each.
- Guns or ammunition of .45 caliber. In the United States, "45" is often a reference to one of two specific .45 caliber cartridges—the .45 Colt or the .45 ACP.
- The number of the French department Loiret.
- The maximum mark an International Baccalaureate student can obtain.
- In years of marriage, the sapphire wedding anniversary.
- Forty-five (audio drama), a Big Finish 2008 audio play made for the forty fifth anniversary of the British science fiction television show Doctor Who.
- Issue 45 of The North Briton was thought to be seditious but its publisher, John Wilkes, was celebrated as a champion of liberty. The number 45 was used as a symbol of support for him. Banquets were held with a theme of 45 while many items were produced showing the number or featuring it in some way. For example, a wig was produced with 45 curls.[15]
- The number of the laps of the German Grand Prix from 1978 until 2001 (except 1985 because the race was held at Nürburgring).
- Donald Trump, 45th President of the United States.
See also
References
