27 (number)

Source: Wikipedia, the free encyclopedia.
← 26 27 28 →
Cardinaltwenty-seven
Ordinal27th
Factorization33
Divisors1, 3, 9, 27
Greek numeralΚΖ´
Roman numeralXXVII
Binary110112
Ternary10003
Senary436
Octal338
Duodecimal2312
Hexadecimal1B16

27 (twenty-seven; Roman numeral XXVII) is the natural number following 26 and preceding 28.

Mathematics

Twenty-seven is the

cube of 3, or three tetrated
, divisible by the number of ).

The first non-trivial decagonal number is 27.[1]

27 has an aliquot sum of 13[2] (the sixth prime number) in the aliquot sequence of only one composite number, rooted in the 13-aliquot tree.[3]

The sum of the first four composite numbers is ,[4] while the sum of the first four prime numbers is ,[5] with 7 the fourth indexed prime.[6][a]

In the Collatz conjecture (i.e. the problem), a starting value of 27 requires 3 × 37 = 111 steps to reach 1, more than any smaller number.[10][b]

27 is also the fourth perfect totient number — as are all powers of 3 — with its adjacent members 15 and 39 adding to twice 27.[13][c]

A prime reciprocal magic square based on multiples of in a square has a magic constant of 27.

Including the null-motif, there are 27 distinct hypergraph motifs.[14]

The Clebsch surface, with 27 straight lines

There are exactly

twenty-seven straight lines on a smooth cubic surface,[15] which give a basis of the fundamental representation of Lie algebra
.[16][17]

The unique simple formally real Jordan algebra, the exceptional Jordan algebra of self-adjoint 3 by 3 matrices of quaternions, is 27-dimensional;[18] its automorphism group is the 52-dimensional exceptional Lie algebra [19]

There are twenty-seven sporadic groups, if the non-strict group of Lie type (with an irreducible representation that is twice that of in 104 dimensions)[20] is included.[21]

In

Robin's theorem for the Riemann hypothesis
, twenty-seven integers fail to hold for values where is the this inequality holds for every larger [22][23][24]

Base-specific

In decimal, 27 is the first composite number not divisible by any of its digits, as well as:

  • the third Smith number[25] and sixteenth Harshad number,[26]
  • the only positive integer that is three times the sum of its digits,
  • equal to the sum of the numbers between and including its digits: .

Also in base ten, if one cyclically rotates the digits of a three-digit number that is a multiple of 27, the new number is also a multiple of 27. For example, 378, 783, and 837 are all divisible by 27.

  • In similar fashion, any multiple of 27 can be mirrored and spaced with a zero each for another multiple of 27 (i.e. 27 and 702, 54 and 405, and 378 and 80703 are all multiples of 27).
  • Any multiple of 27 with "000" or "999" inserted yields another multiple of 27 (20007, 29997, 50004, and 59994 are all multiples of 27).

In senary (base six), one can readily test for divisibility by 43 (decimal 27) by seeing if the last three digits of the number match 000, 043, 130, 213, 300, 343, 430, or 513.

In decimal representation, 27 is located at the twenty-eighth (and twenty-ninth) digit after the decimal point in π:

If one starts counting with zero, 27 is the second self-locating string after 6, of only a few known.[27][28]

In science

Astronomy

Electronics

In language and literature

In astrology

  • 27 Nakṣatra or lunar mansions in Hindu astrology.

In art

Movies

Music

  • "27", a song by Fall Out Boy on the album Folie à Deux
  • "27", a song by Passenger on the album Whispers
  • "27", a song by Title Fight on the album Shed.[34]
  • "27", a song by Biffy Clyro on the album Blackened Sky
  • "27", a song on Machine Gun Kelly's album Bloom
  • "27 Jennifers", a song by Mike Doughty on the album Rockity Roll
  • 27, an album by South Korean singer Kim Sung-kyu
  • 27, an album by Argentine rock band Ciro y los Persas
  • Twenty-Seven, an album by The Adicts.
  • French rapper Kaaris' signature number is 27, from his zip code 93270.
  • "Twenty Seven Strangers" by Villagers.
  • 27, a Boston-based band
  • 27, an opera by composer Ricky Ian Gordon and librettist Royce Vavrek
  • 27 Club, a list of popular musicians, artists, or actors who died at age 27
  • "Weird Al" Yankovic has a recurring joke involving the number 27, which is used in several songs.

Other

  • The Minneapolis-based artist Deuce 7 (a.k.a. Deuce Seven, Twenty Seven, 27).

In sports

  • The value of all the colors in snooker add up to 27.
  • The number of outs in a regulation baseball game for each team at all adult levels, including professional play, is 27.
  • The
    MLB
    .

In other fields

Twenty-seven is also:

  • A-27, American attack aircraft.
  • The code for international direct-dial phone calls to South Africa.
  • The name of a cigarette,
    Marlboro
    Blend No. 27.
  • The number of the French department Eure.

See also

Notes

  1. ^ Whereas the composite index of 27 is 17[7] (the cousin prime to 13),[8] 7 is the prime index of 17.[6]
    The sum  27 + 17 + 7 = 53  represents the sixteenth indexed prime (where 42 = 16).
    While 7 is the fourth prime number, the fourth composite number is 9 = 32, that is also the composite index of 16.[9]
  2. ^ On the other hand,
    • The reduced Collatz sequence of 27, that counts the number of prime numbers in its trajectory, is 41.[11]
      This count represents the thirteenth prime number, that is also in equivalence with the sum of members in the aliquot tree (27, 13, 1, 0).[3][2]
    • The next two larger numbers in the Collatz conjecture to require more than 111 steps to return to 1 are 54 and 55
    • Specifically, the fourteenth prime number 43 requires twenty-seven steps to reach 1.
    The sixth pair of twin primes is (41, 43),[12] whose respective prime indices generate a sum of 27.
  3. ^ Also,  36 = 62  is the sum between PTNs  39 – 15 = 24  and  3 + 9 = 12. In this sequence, 111 is the seventh PTN.

References

  1. ^ "Sloane's A001107 : 10-gonal (or decagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
  2. ^ a b Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
  3. ^ a b Sloane, N. J. A., ed. (January 11, 1975). "Aliquot sequences". Mathematics of Computation. 29 (129). OEIS Foundation: 101–107. Retrieved October 31, 2023.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A151742 (Composite numbers which are the sum of four consecutive composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved November 2, 2023.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A007504 (Sum of the first n primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved November 2, 2023.
  6. ^ a b Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A046132 (Larger member p+4 of cousin primes (p, p+4).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved November 8, 2023.
  10. ^ Sloane, N. J. A. (ed.). "Sequence A112695 (Number of steps needed to reach 4,2,1 in Collatz' 3*n+1 conjecture.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
  11. ^ Sloane, N. J. A. (ed.). "Sequence A286380 (a(n) is the minimum number of iterations of the reduced Collatz function R required to yield 1. The function R (A139391) is defined as R(k) equal to (3k+1)/2^r, with r as large as possible.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved November 8, 2023.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A077800 (List of twin primes {p, p+2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved November 8, 2023.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved November 2, 2023.
  14. S2CID 221779386
    .
  15. ^ Baez, John Carlos (February 15, 2016). "27 Lines on a Cubic Surface". AMS Blogs. American Mathematical Society. Retrieved October 31, 2023.
  16. .
  17. ^ Sloane, N. J. A. (ed.). "Sequence A121737 (Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
  18. .
  19. .
  20. .
  21. .
  22. Zbl 07713337.{{cite journal}}: CS1 maint: Zbl (link
    )
  23. .
  24. ^ Sloane, N. J. A. (ed.). "Sequence A067698 (Positive integers such that sigma(n) is greater than or equal to exp(gamma) * n * log(log(n)).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
  25. ^ "Sloane's A006753 : Smith numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
  26. ^ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved May 31, 2016.
  27. ^ Dave Andersen. "The Pi-Search Page". angio.net. Retrieved October 31, 2023.
  28. ^ Sloane, N. J. A. (ed.). "Sequence A064810 (Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where 1 is the 0th digit.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved October 31, 2023.
  29. ^ "Dark Energy, Dark Matter | Science Mission Directorate". science.nasa.gov. Retrieved November 8, 2020.
  30. ^ "Catalog of Solar Eclipses of Saros 27". NASA Eclipse Website. NASA. Retrieved February 27, 2022.
  31. ^ "Catalog of Lunar Eclipses in Saros 27". NASA Eclipse Website. NASA. Retrieved February 27, 2022.
  32. ^ "SpanishDict Grammar Guide". SpanishDict. Retrieved August 19, 2020.
  33. ^ "Shed, by Title Fight".

Further reading

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987), p. 106.

External links