210 (two hundred ndten) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

following 209 and preceding

211 Year 211 ( CCXI) was a common year starting on Tuesday of the Julian calendar. At the time, in the Roman Empire it was known as the Year of the Consulship of Terentius and Bassus (or, less frequently, year 964 ''Ab urbe condita''). The denomin ...

Mathematics

210 is an

abundant number In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total ...

, and Harshad number. It is the product of the first four

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s ( 2, 3, 5, and 7), and thus a

primorial In mathematics, and more particularly in number theory, primorial, denoted by "", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...

, 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: Wells, D. (1987). ''The Penguin Dictionary of Curious and Interesting Numbers'' (p. 143). London: Penguin Group. It is the 20th

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...

(following 190 and preceding 231), a

pentagonal number A pentagonal number is a figurate number that extends the concept of triangular number, triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotational ...

(following 176 and preceding 247), and the second smallest to be both triangular and pentagonal (the first is 1; the third is 40755). It is also an idoneal number, a pentatope number, a

pronic number A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular number ...

, and an

untouchable number In mathematics, an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back at l ...

. 210 is also the third 71-gonal number, preceding 418. 210 is index in the number of ways to pair up so that the sum of each pair is

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

; i.e., in . 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 .

References

{{Integers, 2 Integers