another proof sequence

[N. J. A. Sloane]

Dear Yasutoshi,   I am adding this sequencx\e:

%I A112404
%S A112404 3,35,75361
%N A112404 a(n) = (Product_{0<=e_i<=1} (Product_{1<=i<=n} p_i^e_i + Product_{1<=i<=n} p_i^(1-e_i)))^(1/2) where p_i means i-th prime.
%C A112404 This is a "Proof of existence of infinite primes" sequence. Proof. Let N = (Product_{0<=e_i<=1} (Product_{1<=i<=n} p_i^e_i + Product_{1<=i<=n} p_i^(1-e_i)))^(1/2) . Suppose there are only a finite number of primes p_i,
1<=i<=n. If N is prime, then for all i, not (N=p_i). Because, for all i, p_i<N. If N is composite, then it must have a prime divisor p which is different from primes p_i. Because, for all i, not (N_1=0, Mod p_i).
%e A112404 a(3)=
%e A112404 ((1+p_1*p_2*p_3)*(p_3+p_1*p_2)*(p_2+p_1*p_3)*(p_2*p_3+p_1)*(p_1+p_2*p_3)*(p_1*p_3+p_2)*(p_1*p_2+p_3)*(p_1*p_2*p_3+1))^(1/2)
%e A112404 =
%e A112404 (1+p_1*p_2*p_3)*(p_3+p_1*p_2)*(p_2+p_1*p_3)*(p_2*p_3+p_1)
%e A112404 = 31*11*13*17
%Y A112404 Cf. A111392.
%K A112404 nonn,uned
%O A112404 1,1
%A A112404 Yasutsohi Kohmoto zbi74583(AT)boat.zero.ad.jp