Sequence of primorial primes
The sequence of primorial primes is given as the increasing values of n where primorial(n) ± 1 is prime.
Noting that the n'th primorial is defined as the multiplication of the smallest n primes, the sequence is of the number of primes, in order that when multiplied together is one-off being a prime number itself.
The task is to generate and show here the first ten values of the sequence. An optional extended task is to show the first twenty members of the series.
Note: this task asks for the primorial indices that create the final primorial prime numbers, so there should be no ten digit decimal numbers in the program output (although extended precision integers will be needed for intermediate results).
- Cf.
- wp:Primorial primes.
- Primorial prime from The Prime Glossary.
- Sequence A088411 from The On-Line Encyclopedia of Integer Sequences
Python
Uses the pure python library pyprimes . Note that pyprimes.isprime switches to a fast and good probabilistic algorithm for larger integers.
<lang python>import pyprimes
def primorial_prime(_pmax=500):
isprime = pyprimes.isprime n, primo = 0, 1 for prime in pyprimes.nprimes(_pmax): n, primo = n+1, primo * prime if isprime(primo-1) or isprime(primo+1): yield n
if __name__ == '__main__':
# Turn off warning on use of probabilistic formula for prime test pyprimes.warn_probably = False for i, n in zip(range(20), primorial_prime()): print('Primorial prime %2i at primorial index: %3i' % (i+1, n))</lang>
- Output:
Primorial prime 1 at primorial index: 1 Primorial prime 2 at primorial index: 2 Primorial prime 3 at primorial index: 3 Primorial prime 4 at primorial index: 4 Primorial prime 5 at primorial index: 5 Primorial prime 6 at primorial index: 6 Primorial prime 7 at primorial index: 11 Primorial prime 8 at primorial index: 13 Primorial prime 9 at primorial index: 24 Primorial prime 10 at primorial index: 66 Primorial prime 11 at primorial index: 68 Primorial prime 12 at primorial index: 75 Primorial prime 13 at primorial index: 167 Primorial prime 14 at primorial index: 171 Primorial prime 15 at primorial index: 172 Primorial prime 16 at primorial index: 287 Primorial prime 17 at primorial index: 310 Primorial prime 18 at primorial index: 352 Primorial prime 19 at primorial index: 384 Primorial prime 20 at primorial index: 457