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%I #19 Feb 16 2020 01:00:57
%S 3,19,11251,2980024297506569894680811251
%N Primes of the form 1 + Product_{k=1..n} prime(k)^(2^(k-1)).
%C Primes of the form 1 + A191554(k), associated with positions k = 1, 2, 3, and 5 there. The next one (if it exists) occurs at k >= 15 and has > 53500 digits. [Edited by _R. J. Mathar_, Jun 17 2011 and _Joerg Arndt_, Jun 21 2011]
%C This connects A191554 and A191555, which are deeply about primes and monic polynomial irreducible by Eisenstein's Criterion, to primes by another way, connecting additive and multiplicative number theory analogously to the relationship in Primorial primes: A014545, n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime.
%e a(1) = 1 + 2^1 = 1 + 2 = 3 is prime.
%e a(2) = 1 + (2^1 * 3^2) = 1 + 18 = 19 is prime.
%e a(3) = 1 + (2^1 * 3^2 * 5^4) = 1 + 11250 = 11251 is prime.
%t Select[Array[1 + Product[Prime[k]^(2^(k - 1)), {k, #}] &, 5], PrimeQ] (* _Michael De Vlieger_, Feb 15 2020 *)
%Y Cf. A000040, A191554.
%K nonn
%O 1,1
%A _Jonathan Vos Post_, Jun 09 2011
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