65536 (number)

← 65535
65537 →
65536
Cardinal sixty-five thousand five hundred thirty-six
Ordinal 65536th
(sixty-five thousand five hundred thirty-sixth)
Factorization 2^{16}
Divisors 17
Roman numeral LXVDXXXVI
Binary 100000000000000002
Octal 2000008
Duodecimal 31B1412
Hexadecimal 1000016

65536 is the natural number following 65535 and preceding 65537.

65536 is a power of two: 2^{16} (2 to the 16th power).

65536 is the smallest number with exactly 17 divisors.[1]

Contents

In mathematics

65536 is 2^{2^{2^{2}}}, so in tetration notation 65536 is ^42.

When expressed using Knuth's up-arrow notation, 65536 is 2 \uparrow 16 , which is equal to 2 \uparrow 2 \uparrow 2 \uparrow 2 , which is equivalent to 2 \uparrow\uparrow 4 or 2 \uparrow\uparrow\uparrow 3 .

65536 is a superperfect number - a number such that σ(σ(n))=2n.[2]

A 16-bit number can distinguish 65536 different possibilities, such as the numbers 0..65535. In unsigned binary notation, 65536 is thus the smallest number that can not be represented by 16 bits.

65536 is the only power of 2 less than 231000 that does not contain the digits 1, 2, 4 or 8 in its decimal representation.[3]

65536 is the largest known number such that the sum of its unitary divisors is prime (1 + 65536 = 65537, which is prime).[4]

In computing

65536 (216) is the number of different values representable in a number of 16 binary digits, also known as a short integer in many computer programming systems. Therefore,

In popular culture

There are 65536 different charts in Western geomancy.

References

  1. ^ Sequence  A005179 in OEIS
  2. ^ Weisstein, Eric W., "Superperfect Number" from MathWorld.
  3. ^ Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers (revised ed.). Penguin. ISBN 0-14-026149-4. 
  4. ^ 65536 at Prime Pages
  5. ^ Microsoft Help Q120596