45 (number)

← 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

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 , 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, and first of the form . 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]

Abstract algebra

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 . Inside the largest sporadic group, the Friendly Giant , there exist at least 45 conjugacy classes of maximal subgroups, which includes the double cover of the second largest sporadic group .[14]

In science

Astronomy

In music

45 rpm gramophone record

In other fields

Forty-five may also refer to:

See also

References

  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. ^ Dietrich, Heiko; Lee, Melissa; Popiel, Tomasz (May 2023). "The maximal subgroups of the Monster": 1–11. arXiv:2304.14646. S2CID 258676651. {cite journal}: Cite journal requires |journal= (help)
  15. ^ Arthur Hill Cash (2007), John Wilkes: The Scandalous Father of Civil Liberty, Yale University Press, p. 219, ISBN 978-0-300-12363-0