45 (number)

Short description: Natural number

| <- 44  | 45 |  46 ->

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← 40 41 42 43 44 45 46 47 48 49 →

List of numbers — Integers

← 0 10 20 30 40 50 60 70 80 90 →

Cardinal| forty-five
Ordinal| 45th
(forty-fifth)
Factorization| 32 × 5
Divisors| 1, 3, 5, 9, 15, 45
Greek numeral| ΜΕ´
Roman numeral| XLV
Binary| 1011012
Ternary| 12003
Quaternary| 2314
Quinary| 1405
Senary| 1136
Octal| 558
Duodecimal| 3912
Hexadecimal| 2D16
Vigesimal| 2520
Base 36| 1936

45 (forty-five) is the natural number following 44 and preceding 46.

## Contents

* 1 In mathematics
* 1.1 Abstract algebra
* 2 In science
* 2.1 Astronomy
* 3 In music
* 4 In other fields
* 5 See also
* 6 References

## In mathematics

45 as the difference of two nonzero squares (in orange)

Forty-five is the smallest odd number that has more divisors than n+1, and that has a larger sum of divisors than n+1.[1][2] It is the sixth positive integer with a square-prime prime factorization of the form p2q, with p and q prime, and first of the form 32×q. 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: 0+1+2+3+4+5+6+7+8+9=45. 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: 72−22, 92−62 or 232−222 (see image).[6]

Since the greatest prime factor of 452+1=2026 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 R(5,5).[11][12]

ϕ(45)=ϕ(σ(45))[13]

### Abstract algebra

In the classification of finite simple groups, the Tits group 𝕋 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 𝕋.

## 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

45 rpm gramophone record

* 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
* forty-five is repeated continuously in the lyrics of the 1997 song "Brimful of Asha" by Cornershop

## In other fields

See also: List of highways numbered 45

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.[14]
* 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

* List of highways numbered 45

## References

1. ↑ Sloane, N. J. A., ed. "Sequence A138171". OEIS Foundation. https://oeis.org/A138171. Retrieved 2022-11-25.
2. ↑ Sloane, N. J. A., ed. "Sequence A067828". OEIS Foundation. https://oeis.org/A067828. Retrieved 2022-11-25.
3. ↑ Sloane, N. J. A., ed. "Sequence A000217 (Triangular numbers)". OEIS Foundation. https://oeis.org/A000217. Retrieved 2016-05-30.
4. ↑ Sloane, N. J. A., ed. "Sequence A000384 (Hexagonal numbers)". OEIS Foundation. https://oeis.org/A000384. Retrieved 2016-05-30.
5. ↑ Sloane, N. J. A., ed. "Sequence A051868 (16-gonal numbers)". OEIS Foundation. https://oeis.org/A051868. Retrieved 2016-05-30.
6. ↑ (sequence A334078 in the OEIS)
7. ↑ Sloane, N. J. A., ed. "Sequence A005528 (Størmer numbers)". OEIS Foundation. https://oeis.org/A005528. Retrieved 2016-05-30.
8. ↑ Sloane, N. J. A., ed. "Sequence A006886 (Kaprekar numbers)". OEIS Foundation. https://oeis.org/A006886. Retrieved 2016-05-30.
9. ↑ Sloane, N. J. A., ed. "Sequence A005349 (Niven (or Harshad) numbers)". OEIS Foundation. https://oeis.org/A005349. Retrieved 2016-05-30.
10. ↑ Sloane, N. J. A., ed. "Sequence A001003 (Schroeder's second problem; ... also called super-Catalan numbers or little Schroeder numbers.)". OEIS Foundation. https://oeis.org/A001003. Retrieved 2022-11-25.
11. ↑ Sloane, N. J. A., ed. "Sequence A120414 (Conjectured Ramsey number R(n,n).)". OEIS Foundation. https://oeis.org/A120414. Retrieved 2023-02-17.
12. ↑ Sloane, N. J. A., ed. "Sequence A212954 (Triangle read by rows: two color Ramsey numbers)". OEIS Foundation. https://oeis.org/A212954. Retrieved 2022-11-25.
13. ↑ Sloane, N. J. A., ed. "Sequence A006872". OEIS Foundation. https://oeis.org/A006872.
14. ↑ Arthur Hill Cash (2007), John Wilkes: The Scandalous Father of Civil Liberty, Yale University Press, p. 219, ISBN 978-0-300-12363-0, https://archive.org/details/johnwilkes00arth_0/page/219

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