Natural number
45 (forty-five ) is the
natural number following
44 and preceding
46 .
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
{\displaystyle n+1}
, and that has a larger sum of divisors than
n
+
1
{\displaystyle n+1}
.
[1]
[2] It is the sixth
positive integer with a square-prime
prime factorization of the form
p
2
q
{\displaystyle p^{2}q}
, with
p
{\displaystyle p}
and
q
{\displaystyle q}
prime . 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
{\displaystyle 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:
7
2
−
2
2
{\displaystyle 7^{2}-2^{2}}
,
9
2
−
6
2
{\displaystyle 9^{2}-6^{2}}
or
23
2
−
22
2
{\displaystyle 23^{2}-22^{2}}
(see image).
[6]
Since the greatest prime factor of
45
2
+
1
=
2026
{\displaystyle 45^{2}+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
)
{\displaystyle R(5,5)}
.
[11]
[12]
ϕ
(
45
)
=
ϕ
(
σ
(
45
)
)
{\displaystyle \phi (45)=\phi (\sigma (45))}
[13]
Forty-five degrees is half of a right angle (90°).
Abstract algebra
In the
classification of finite simple groups , the
Tits group
T
{\displaystyle \mathbb {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
{\displaystyle \mathbb {T} }
.
In science
Astronomy
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
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
References
^
Sloane, N. J. A. (ed.).
"Sequence A138171" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-11-25 .
^
Sloane, N. J. A. (ed.).
"Sequence A067828" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-11-25 .
^
Sloane, N. J. A. (ed.).
"Sequence A000217 (Triangular numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^
Sloane, N. J. A. (ed.).
"Sequence A000384 (Hexagonal numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^
Sloane, N. J. A. (ed.).
"Sequence A051868 (16-gonal numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^ (sequence
A334078 in the
OEIS )
^
Sloane, N. J. A. (ed.).
"Sequence A005528 (Størmer numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^
Sloane, N. J. A. (ed.).
"Sequence A006886 (Kaprekar numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^
Sloane, N. J. A. (ed.).
"Sequence A005349 (Niven (or Harshad) numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-30 .
^
Sloane, N. J. A. (ed.).
"Sequence A001003 (Schroeder's second problem; ... also called super-Catalan numbers or little Schroeder numbers.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-11-25 .
^
Sloane, N. J. A. (ed.).
"Sequence A120414 (Conjectured Ramsey number R(n,n).)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-02-17 .
^
Sloane, N. J. A. (ed.).
"Sequence A212954 (Triangle read by rows: two color Ramsey numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-11-25 .
^
Sloane, N. J. A. (ed.).
"Sequence A006872" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^ Arthur Hill Cash (2007),
John Wilkes: The Scandalous Father of Civil Liberty , Yale University Press, p.
219 ,
ISBN
978-0-300-12363-0
100,000
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100,000,000
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