Sequence of primorial primes

Sequence of primorial primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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-or-more digit 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

Haskell

<lang Haskell> import Data.List (scanl1, findIndices, nub)

primes :: [Integer] primes = 2 : filter isPrime [3,5..]

isPrime :: Integer -> Bool isPrime = isPrime_ primes

 where isPrime_ :: [Integer] -> Integer -> Bool
       isPrime_ (p:ps) n
         | p * p > n      = True
         | n `mod` p == 0 = False
         | otherwise      = isPrime_ ps n

primorials :: [Integer] primorials = 1 : scanl1 (*) primes

primorialsPlusMinusOne :: [Integer] primorialsPlusMinusOne = concatMap (\n -> [n-1, n+1]) primorials

sequenceOfPrimorialPrimes :: [Int] sequenceOfPrimorialPrimes = nub $ map (`div` 2) $ findIndices (==True) bools

 where bools = map isPrime primorialsPlusMinusOne

main :: IO () main = mapM_ print $ take 10 sequenceOfPrimorialPrimes </lang>

Output:

0
1
2
3
4
5
6
11
13
24

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