A208936 Prime production length of the polynomial P = x^2 + x + prime(n): max { k>0 | P(x) is prime for all x=0,...,k-1 }.

1, 2, 4, 1, 10, 1, 16, 1, 1, 2, 1, 1, 40, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1

Offset: 1

Author

M. F. Hasler, Mar 03 2012

Keywords

• nonn

Comments

a(n) > 0 by definition, and a(n) > 1 iff n is a twin prime; a(n) would be zero for composite n if "prime(n)" was replaced by n.

Euler's original "prime producing polynomial" was P = x^2 - x + 41; changing the sign increases the prime production length by 1.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Shaoji Xu, On Euler Polynomials and Rabinowitsch Polynomials, International Journal of Algebra, Vol. 6, 2012, no. 3, 123 - 134.

Programs

Maple

A208936:= proc(n) local N,r;
   N:= ithprime(n);
   for r from 1 do
     if not isprime(r^2+r+N) then return(r) end if
   end do
end proc; # Robert Israel, Feb 11 2013

PARI

a(n)={n=prime(n); for( x=1, 1e9, isprime(x^2+x+n) || return(x))}