

A210482


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


1




OFFSET

1,1


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 ith Euclid number. p = p_i is the ith 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


LINKS



FORMULA



EXAMPLE

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.


PROG

(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)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



