1,1

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) * (Sum_{1<=i<=n} (1/p_i*Product_{1<=k<=n} p_k) ) . 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=0, Mod p_i).

a(3)=

((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)

* (p_2*p_3+p_1*p_3+p_1*p_2)

=

(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_2*p_3+p_1*p_3+p_1*p_2)

= 31*11*13*17*31

Cf. A111392.

Sequence in context: A053930 A053920 A125711 this_sequence A091324 A093434 A053291

Adjacent sequences: A113267 A113268 A113269 this_sequence A113271 A113272 A113273

nonn

Yasutoshi Kohmoto zbi74583(AT)boat.zero.ad.jp, Jan 07 2006