81 (number)

81 (eighty-one) is the natural number following 80 and preceding 82.

In mathematics
81 is:


 * the square of 9 and the second fourth-power of a prime; 34.
 * with an aliquot sum of 40; within an aliquot sequence of three composite numbers (81,40,50,43,1,0) to the Prime in the 43-aliquot tree.
 * a perfect totient number like all powers of three.
 * a heptagonal number.
 * an icosioctagonal number.
 * a centered octagonal number.
 * a tribonacci number.
 * an open meandric number.
 * the ninth member of the Mian-Chowla sequence.
 * a palindromic number in bases 8 (1218) and 26 (3326).
 * a Harshad number in bases 2, 3, 4, 7, 9, 10 and 13.
 * one of three non-trivial numbers (the other two are 1458 and 1729) which, when its digits (in decimal) are added together, produces a sum which, when multiplied by its reversed self, yields the original number:
 * 8 + 1 = 9
 * 9 &times; 9 = 81 (although this case is somewhat degenerate, as the sum has only a single digit).

The inverse of 81 is 0.$\overline{012345679}$ recurring, missing only the digit "8" from the complete set of digits. This is an example of the general rule that, in base b,
 * $$\frac{1}{\left(b-1\right)^2} = 0.\overline{012\cdots(b-4)(b-3)(b-1)},$$

omitting only the digit b&minus;2.

In astronomy

 * Messier object M81, a magnitude 8.5 spiral galaxy in the constellation Ursa Major, also known as Bode's Galaxy, and the first of what is known as the M81 Group of galaxies
 * The New General Catalogue object NGC 81, a spiral galaxy in the constellation Andromeda

In other fields
Eighty-one is also:
 * The number of squares on a shogi playing board
 * The year AD 81, 81 BC, or 1981.
 * The atomic number of thallium
 * The symbolic number of the Hells Angels Motorcycle Club. 'H' and 'A' are the 8th and 1st letter of the alphabet, respectively.
 * The title of a short film by Stephen Burke: 81
 * The model number of Sinclair ZX81
 * The number of the department in France called Tarn [[Image:Tarn-Position.png|thumb|90px|[[Departments of France|Department]] 81 of France (Tarn)]]
 * The code for international direct dial phone calls to Japan
 * "+81" is a song by Japanese metalcore band Crystal Lake.
 * One of two ISBN Group Identifiers for books published in India
 * The number of stanzas or chapters in the Tao Te Ching (in the most common arrangements).
 * The number of provinces in Turkey.
 * The number of prayers said in the Rosary in each night.
 * "The 81" is a 1965 song by Candy and the Kisses.
 * Artemis 81 is a 1981 BBC TV science fiction drama.
 * 'The Eighty-One Brothers' is a Japanese fable
 * The number of possible divinations in the Taixuanjing
 * 81 (八一) is referenced in flags and symbols of the People's Liberation Army as it was founded on August 1 (8/1).