Tool to compute a primorial. Primorial n# is the product of all prime numbers inferior or equal to n, or the product of all n first prime numbers (it depend on the selected definition)
Primorial - dCode
Tag(s) : Arithmetics
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Primorial
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Tool to compute a primorial. Primorial n# is the product of all prime numbers inferior or equal to n, or the product of all n first prime numbers (it depend on the selected definition)
Answers to Questions
What is a primorial? (Definition)
The term primorial refers to two separate definitions/formulae according to some uses:
(1) Primorial defined as the product of all prime numbers inferior or equal to $ n $ is a multiplication conditioned par a primality test of the numbers inferior or equal to $ n $, see OEIS here (link)
Example: $ 6\# = 2 \times 3 \times 5 = 30 $
(2) Primorial defined as a product of the $ n $ first primes is equivalent to a multiplication of the list of the first $ n $ prime numbers, see OEIS here (link)
Example: $ 4\# = 2 \times 3 \times 5 \times 7 = 210 $
The primorial of p is written with the character sharp: p# or $ p\# $
By convention $ 1\# = 1 $
What is the primorial function?
The primorial function is the function that at a natural integer $ n $ associates the value $ n\# $
How to calculate a primorial?
The primorial calculation is a succession of multiplication of prime numbers. According to definitions (1) and (2):
Example:
| n | n# (1) | n# (2) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 2 |
| 3 | 6 | 6 |
| 4 | 6 | 30 |
| 5 | 30 | 210 |
| 6 | 30 | 2310 |
| 7 | 210 | 30030 |
| 8 | 210 | 510510 |
| 9 | 210 | 9699690 |
| 10 | 210 | 223092870 |
| 11 | 2310 | 6469693230 |
| ... | ... | ... |
Lists (1) and (2) contain the same numbers but (1) have repeated elements.
Source code
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