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%I #64 Jul 26 2022 21:59:13
%S 2,5,37,44101,5336101,94083986096101,
%T 1062053250251407755176413469419400772901
%N Prime numbers of the form (p#)^2 + 1, where p# is a primorial.
%C This sequence is the subset of primes in A189409.
%C The sequence A189409 is an extension of Euclid's second theorem about generating infinitely many prime numbers.
%C 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.
%C 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.
%C The next term is about 2.519... * 10^1327. - _Amiram Eldar_, Nov 23 2018
%F Intersection of A000040 and A189409.
%F a(n) = A189409(A092061(n)). - _Amiram Eldar_, Nov 23 2018
%e 2, 5 and 37 of A189409 are primes and in the sequence.
%e But 901=17*53, the next term of A189409, is not a prime and not in the sequence.
%o (Python)
%o from functools import reduce
%o import numpy as np
%o def factors(n):
%o return reduce(list.__add__, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
%o mul=1
%o for i in range(1,20):
%o if len(factors(i))<3:
%o mul*= i*i
%o if len(factors(mul+1))< 3:
%o print(mul+1)
%Y Cf. A000040, A001248, A002110, A061742, A092061, A189409.
%K nonn
%O 1,1
%A _Abhiram R Devesh_, Jan 23 2013
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