A111392 a(n) = Product_{i=1..n-1} (Product_{k=1..i} p_k + Product_{k=i+1..n} p_k).

2, 5, 187, 162319, 10697595389, 63619487169453143, 74365399061678006800073593, 11864736003419293844093922527852416537, 601642845102734414280661105098046392912578705726003

Offset

1,1

Comments

This sequence gives another proof that there are infinitely many primes. Let N = Product_{1<=i<n} (Product_{1<=k<=i} p_k + Product_{i<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(1) could also be chosen to be 1.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..29

Maple

a:=n->mul(mul(ithprime(k), k=1..i)+mul(ithprime(k), k=i+1..n), i=1..n-1): 2, seq(a(n), n=2..10); # Muniru A Asiru, Dec 06 2018

Mathematica

Join[{2}, Rest[f[n_]:=Product[(Product[Prime[k], {k, i}] + Product[Prime[k], {k, i + 1, n}]), {i, n - 1}]; Array[f, 10] ]] (* Robert G. Wilson v, Nov 12 2005 *)

Prog

(PARI) t=10; for(n=2, t, print1(prod(i=1, n-1, prod(k=1, i, prime(k)) + prod(k=i+1, n, prime(k))), ", ")); \\ Gerald McGarvey, Nov 12 2005

Crossrefs

Cf. A024451, A112404.

Sequence in context: A116629 A124275 A013130 * A226071 A319143 A100366

Adjacent sequences: A111389 A111390 A111391 * A111393 A111394 A111395

Keyword

nonn,nice

Author

Yasutoshi Kohmoto, Nov 08 2005

Extensions

Corrected and extended by Gerald McGarvey and Robert G. Wilson v, Nov 12 2005

Status

approved