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A210482 Prime numbers of the form (p#)^2+1, where p# is a primorial. 1
2, 5, 37, 44101, 5336101, 94083986096101, 1062053250251407755176413469419400772901 (list; graph; refs; listen; history; text; internal format)



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 of this series 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


Table of n, a(n) for n=1..7.


Intersection of A000040 and A189409.

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


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.



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))


for i in range(1, 20):

    if len(factors(i))<3:

        mul*= i*i

        if len(factors(mul+1))< 3:



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

Sequence in context: A189409 A222318 A084436 * A053609 A036780 A051501

Adjacent sequences:  A210479 A210480 A210481 * A210483 A210484 A210485




Abhiram R Devesh, Jan 23 2013



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Last modified September 22 13:34 EDT 2021. Contains 347607 sequences. (Running on oeis4.)