A191554 - OEIS

%I #13 Jan 03 2021 17:30:31

%S 2,18,11250,64854011250,2980024297506569894680811250,

%T 1319492964487055911863581348741902326769016593763234907139211250

%N a(n) = Product_{k=1..n} prime(k)^(2^(k-1)).

%C x^(2^n) - a(n) is the minimal polynomial over Q for the algebraic number sqrt(p(n)*sqrt(p(n-1)*...*sqrt(p(2)*sqrt(p(1)))...)), where p(k) is the k-th prime.  Each such monic polynomial is irreducible by Eisenstein's Criterion (using p = 2).

%F For n > 1, a(n) = a(n-1) * prime(n)^(2^(n-1)); a(1) = prime(1).

%e a(1) = 2^1 = 2 and x^2 - 2 is the minimal polynomial for the algebraic number sqrt(2).

%e a(4) = 2^1*3^2*5^4*7^8 = 64854011250 and x^16 - 64854011250 is the minimal polynomial for the algebraic number sqrt(7*sqrt(5*sqrt(3*sqrt(2)))).

%o (PARI) a(n) = prod(k=1, n, prime(k)^(2^(k-1)))

%Y Cf. A191555.

%K nonn,easy

%O 1,1

%A _Rick L. Shepherd_, Jun 05 2011