220 (number)

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

← 219 220 221 →
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Cardinaltwo hundred twenty
Ordinal220th
(two hundred twentieth)
Factorization22 × 5 × 11
Greek numeralΣΚ´
Roman numeralCCXX
Binary110111002
Ternary220113
Octal3348
Duodecimal16412
HexadecimalDC16

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]

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, N. J. A. (ed.). "Sequence A005153 (Practical numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  4. Sloane, N. J. A. (ed.). "Sequence 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, N. J. A. (ed.). "Sequence A034963 (Sums of four consecutive primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. Sloane, N. J. A. (ed.). "Sequence A025018 (Numbers n such that least prime in Goldbach partition of n increases)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. Sloane, N. J. A. (ed.). "Sequence A037444 (Number of partitions of n^2 into squares)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  8. Sloane, N. J. A. (ed.). "Sequence A000292 (Tetrahedral (or triangular pyramidal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  9. Sloane, N. J. A. (ed.). "Sequence A006566 (Dodecahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  10. Sloane, N. J. A. (ed.). "Sequence A007678 (Number of regions in regular n-gon with all diagonals drawn)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  11. Sloane, N. J. A. (ed.). "Sequence 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
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