210 (number)

← 209 210 211 →
Cardinaltwo hundred ten
Ordinal210th
(two hundred tenth)
Factorization2 × 3 × 5 × 7
Divisors1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
Greek numeralΣΙ´
Roman numeralCCX, ccx
Binary110100102
Ternary212103
Senary5506
Octal3228
Duodecimal15612
HexadecimalD216

210 (two hundred [and] ten) is the natural number following 209 and preceding 211.

Mathematics

210 is an abundant number,[1] and Harshad number. It is the product of the first four prime numbers (2, 3, 5, and 7), and thus a primorial,[2] where it is the least common multiple of these four prime numbers. 210 is the first primorial number greater than 2 which is not adjacent to 2 primes (211 is prime, but 209 is not).

It is the sum of eight consecutive prime numbers, between 13 and the thirteenth prime number: 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210.[3]

It is the 20th triangular number (following 190 and preceding 231),[4] a pentagonal number (following 176 and preceding 247), and the second smallest to be both triangular and pentagonal (the third is 40755).[3]

It is also an idoneal number, a pentatope number, a pronic number, and an untouchable number. 210 is also the third 71-gonal number, preceding 418.[3]

210 is index n = 7 in the number of ways to pair up {1, ..., 2n} so that the sum of each pair is prime; i.e., in {1, ..., 14}.[5][6]

It is the largest number n where the number of distinct representations of n as the sum of two primes is at most the number of primes in the interval [⁠n/2 ⁠, n − 2].[7]

See also

References

  1. ^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-10.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A002110 (Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-10.
  3. ^ a b c Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers (p. 143). London: Penguin Group.
  4. ^ "A000217 - OEIS". oeis.org. Retrieved 2024-11-28.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A000341 (Number of ways to pair up {1..2n} so sum of each pair is prime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-10.
  6. ^ Greenfield, Lawrence E.; Greenfield, Stephen J. (1998). "Some Problems of Combinatorial Number Theory Related to Bertrand's Postulate". Journal of Integer Sequences. 1. Waterloo, ON: David R. Cheriton School of Computer Science: Article 98.1.2. MR 1677070. S2CID 230430995. Zbl 1010.11007.
  7. ^ Deshouillers, Jean-Marc; Granville, Andrew; Narkiewicz, Władysław; Pomerance, Carl (1993). "An upper bound in Goldbach's problem". Mathematics of Computation. 61 (203): 209–213. Bibcode:1993MaCom..61..209D. doi:10.1090/S0025-5718-1993-1202609-9.