210 (number)
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|---|---|---|---|---|
| Cardinal | two hundred ten | |||
| Ordinal | 210th (two hundred tenth) | |||
| Factorization | 2 × 3 × 5 × 7 | |||
| Divisors | 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210 | |||
| Greek numeral | ΣΙ´ | |||
| Roman numeral | CCX, ccx | |||
| Binary | 110100102 | |||
| Ternary | 212103 | |||
| Senary | 5506 | |||
| Octal | 3228 | |||
| Duodecimal | 15612 | |||
| Hexadecimal | D216 | |||
210 (two hundred [and] ten) is the natural number following 209 and preceding 211.
Mathematics
[edit]210 is an abundant number,[1] and Harshad number.[2] It is the product of the first four prime numbers (2, 3, 5, and 7), and thus a primorial,[3] 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.[4]
It is the 20th triangular number (following 190 and preceding 231),[5] a pentagonal number (following 176 and preceding 247),[6] and the second smallest to be both triangular and pentagonal (the first is 1; the third is 40755).[4] It is also a pentatope number.[7]
It is also an idoneal number,[8] a pronic number,[9] and an untouchable number.
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}.[10][11]
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].[12]
References
[edit]- ↑ 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.
- ↑ "A005349 - OEIS". oeis.org. Retrieved 2026-06-17.
- ↑ 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.
- 1 2 Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers (p. 143). London: Penguin Group.
- ↑ "A000217 - OEIS". oeis.org. Retrieved 2024-11-28.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000326 (pentagonal number)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ "A000332 - OEIS". oeis.org. Retrieved 2026-06-17.
- ↑ "A000926 - OEIS". oeis.org. Retrieved 2026-06-17.
- ↑ "A002378 - OEIS". oeis.org. Retrieved 2026-06-17.
- ↑ 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.
- ↑ 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.
- ↑ 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.
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