← 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 prime numbers below it (nine).

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 Euler–Mascheroni constant; this hypothesis is true if and only if 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:

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 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

In sports

In other fields

Twenty-seven is also:

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. ^ Lee, Geon; Ko, Jihoon; Shin, Kijung (2020). "Hypergraph Motifs: Concepts, Algorithms, and Discoveries". In Balazinska, Magdalena; Zhou, Xiaofang (eds.). 46th International Conference on Very Large Data Bases. Proceedings of the VLDB Endowment. Vol. 13. ACM Digital Library. pp. 2256–2269. arXiv:2003.01853. doi:10.14778/3407790.3407823. ISBN 9781713816126. OCLC 1246551346. 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. ^ Aschbacher, Michael (1987). "The 27-dimensional module for E6. I". Inventiones Mathematicae. 89. Heidelberg, DE: Springer: 166–172. Bibcode:1987InMat..89..159A. doi:10.1007/BF01404676. MR 0892190. S2CID 121262085. Zbl 0629.20018.
  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. ^ Kac, Victor Grigorievich (1977). "Classification of Simple Z-Graded Lie Superalgebras and Simple Jordan Superalgebras". Communications in Algebra. 5 (13). Taylor & Francis: 1380. doi:10.1080/00927877708822224. MR 0498755. S2CID 122274196. Zbl 0367.17007.
  19. ^ Baez, John Carlos (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2). Providence, RI: American Mathematical Society: 189–191. doi:10.1090/S0273-0979-01-00934-X. MR 1886087. S2CID 586512. Zbl 1026.17001.
  20. ^ Lubeck, Frank (2001). "Smallest degrees of representations of exceptional groups of Lie type". Communications in Algebra. 29 (5). Philadelphia, PA: Taylor & Francis: 2151. doi:10.1081/AGB-100002175. MR 1837968. S2CID 122060727. Zbl 1004.20003.
  21. ^ Hartley, Michael I.; Hulpke, Alexander (2010). "Polytopes Derived from Sporadic Simple Groups". Contributions to Discrete Mathematics. 5 (2). Alberta, CA: University of Calgary Department of Mathematics and Statistics: 27. doi:10.11575/cdm.v5i2.61945. ISSN 1715-0868. MR 2791293. S2CID 40845205. Zbl 1320.51021.
  22. ^ Axler, Christian (2023). "On Robin's inequality". The Ramanujan Journal. 61 (3). Heidelberg, GE: Springer: 909–919. arXiv:2110.13478. Bibcode:2021arXiv211013478A. doi:10.1007/s11139-022-00683-0. S2CID 239885788. Zbl 1532.11010.
  23. ^ Robin, Guy (1984). "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann" (PDF). Journal de Mathématiques Pures et Appliquées. Neuvième Série (in French). 63 (2): 187–213. ISSN 0021-7824. MR 0774171. Zbl 0516.10036.
  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. ^ Steve Jenkins, Bones (2010), ISBN 978-0-545-04651-0
  31. ^ "Catalog of Solar Eclipses of Saros 27". NASA Eclipse Website. NASA. Retrieved February 27, 2022.
  32. ^ "Catalog of Lunar Eclipses in Saros 27". NASA Eclipse Website. NASA. Retrieved February 27, 2022.
  33. ^ "SpanishDict Grammar Guide". SpanishDict. Retrieved August 19, 2020.

Further reading

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