220 (number)

220 (two hundred [and] twenty) is the natural number following 219 and preceding 221.

In mathematics

It is a composite number, with its divisors being 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, making it an amicable number with 284.^[1][2] Every number up to 220 may be expressed as a sum of its divisors, making 220 a practical number.^[3] Also, being divisible by the sum of its digits, 220 is a Harshad number.^[4]

It is the sum of four consecutive primes (47 + 53 + 59 + 61).^[5] It is the smallest even number with the property that when represented as a sum of two prime numbers (per Goldbach's conjecture) both of the primes must be greater than or equal to 23.^[6] There are exactly 220 different ways of partitioning 64 = 8² into a sum of square numbers.^[7]

It is a tetrahedral number, the sum of the first ten triangular numbers,^[8] and a dodecahedral number.^[9] If all of the diagonals of a regular decagon are drawn, the resulting figure will have exactly 220 regions.^[10]

It is the sum of the sums of the divisors of the first 16 positive integers.^[11]

In other fields

The number 220 can also refer to:

• The years 220 AD and 220 BC.
• A common voltage in many countries.
• An SMTP code meaning "service ready"
• A series of garden tractors produced by Case (and later by Case IH) in the 1970s and 1980s - the series included the 220(10HP), the 222(12HP) and the 224(14HP).
• 220 film, a film format for medium format cameras.
• A single by music duo t.A.T.u.
• An album by guitarist Phil Keaggy.
• The InterCity 225 train, a newer version of the InterCity 125.
• .220 Russian and .220 Swift, firearm cartridges

Notes

1. ↑ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 167
2. ↑ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 61. ISBN 978-1-84800-000-1. 
3. ↑ "Sloane's A005153 : Practical numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
4. ↑ "Sloane's A005349 : Niven (or Harshad) numbers: numbers that are divisible by the sum of their digits". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
5. ↑ "Sloane's A034963 : Sums of four consecutive primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
6. ↑ "Sloane's A025018 : Numbers n such that least prime in Goldbach partition of n increases". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
7. ↑ "Sloane's A037444 : Number of partitions of n^2 into squares". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
8. ↑ "Sloane's A000292 : Tetrahedral (or triangular pyramidal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
9. ↑ "Sloane's A006566 : Dodecahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
10. ↑ "Sloane's A007678 : Number of regions in regular n-gon with all diagonals drawn". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
11. ↑ "Sloane's A024916 : sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 

References

• Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers (pp. 145 – 147). London: Penguin Group.