45 (number)

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

← 44 45 46 →
Cardinalforty-five
Ordinal45th
(forty-fifth)
Factorization32 × 5
Divisors1, 3, 5, 9, 15, 45
Greek numeralΜΕ´
Roman numeralXLV
Binary1011012
Ternary12003
Senary1136
Octal558
Duodecimal3912
Hexadecimal2D16

In mathematics

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

Forty-five is the smallest odd number that has more divisors than  , and that has a larger sum of divisors than  .[1][2] It is the sixth positive integer with a square-prime prime factorization of the form  , with   and   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:  . 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:  ,   or   (see image).[6]

Since the greatest prime factor of   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  .[11][12]

 [13]

Forty-five degrees is half of a right angle (90°).

Abstract algebra

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

In science

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Astronomy

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In music

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45 rpm gramophone record

In other fields

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Forty-five may also refer to:

See also

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References

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  1. ^ Sloane, N. J. A. (ed.). "Sequence A138171". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A067828". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A000384 (Hexagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A051868 (16-gonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  6. ^ (sequence A334078 in the OEIS)
  7. ^ Sloane, N. J. A. (ed.). "Sequence A005528 (Størmer numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A006886 (Kaprekar numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 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.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-25.
  11. ^ 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.
  12. ^ 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.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A006872". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  14. ^ Arthur Hill Cash (2007), John Wilkes: The Scandalous Father of Civil Liberty, Yale University Press, p. 219, ISBN 978-0-300-12363-0