A210482 Prime numbers of the form (p#)^2 + 1, where p# is a primorial.

2, 5, 37, 44101, 5336101, 94083986096101, 1062053250251407755176413469419400772901

Offset: 1

Author

Abhiram R Devesh, Jan 23 2013

Keywords

• nonn

Comments

This sequence is the subset of primes in A189409.

The sequence A189409 is an extension of Euclid's second theorem about generating infinitely many prime numbers.

The motivation for this sequence is Euclid's second theorem or infinitude of primes theorem. Per this theorem, N = (2*3*5..p) + 1 generates the i-th Euclid number. p = p_i is the i-th prime. This Euclid number is either a prime or product of primes with one of the prime factors greater than p_i.

This is generated as a product of the squares of the first N prime numbers and adding 1 to it. M = ((2*2)*(3*3)*(5*5)*...*(p*p)) + 1. a(8) is a possible prime of 1328 digits.

The next term is about 2.519... * 10^1327. - Amiram Eldar, Nov 23 2018

Examples

			2, 5 and 37 of A189409 are primes and in the sequence.
But 901=17*53, the next term of A189409, is not a prime and not in the sequence.
		

Crossrefs

Cf. A000040, A001248, A002110, A061742, A092061, A189409.

Programs

Python

from functools import reduce
import numpy as np
def factors(n):
    return reduce(list._add_, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
mul=1
for i in range(1,20):
    if len(factors(i))<3:
        mul*= i*i
        if len(factors(mul+1))< 3:
            print(mul+1)

Formula

Intersection of A000040 and A189409.

a(n) = A189409(A092061(n)). - Amiram Eldar, Nov 23 2018