A239146 Smallest k>0 such that n +/- k and n^2 +/- k are all prime. a(n) = 0 if no such number exists.

0, 0, 0, 0, 0, 0, 0, 3, 2, 3, 0, 5, 0, 3, 2, 0, 0, 13, 12, 0, 2, 0, 0, 0, 6, 15, 10, 0, 12, 0, 0, 15, 20, 0, 12, 5, 0, 15, 22, 21, 12, 0, 0, 0, 14, 27, 0, 35, 0, 0, 8, 15, 0, 0, 24, 27, 0, 0, 48, 7, 48, 0, 50, 3, 6, 7, 0, 0, 28, 0, 18, 0, 0, 27, 34

Offset: 1

Author

Derek Orr, Mar 11 2014

Keywords

• nonn

Comments

a(n) is always smaller than n due to the requirement on n-k.

Examples

			8 +/- 1 (7 and 9) and 8^2 +/- 1 (63 and 65) are not all prime. 8 +/- 2 (6 and 10) and 8^2 +/- 2 (62 and 66) are not all prime. However, 8 +/- 3 (5 and 11) and 8^2 +/- 3 (61 and 67) are all prime. Thus, a(8) = 3.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A082467, A172990, A172989.

Programs

Maple

A239146 := proc(n)
    local k ;
    for k from 1 do
        if n-k <= 1 then
            return 0;
        end if;
        if isprime(n+k) and isprime(n-k) and isprime(n^2+k)
            and isprime(n^2-k) then
            return k;
        end if;
    end do;
end proc:
seq(A239146(n),n=1..80) ; # R. J. Mathar, Mar 12 2014

Mathematica

a[n_] := Catch@ Block[{k = 1}, While[k < n, And @@ PrimeQ@ {n+k, n-k, n^2+k, n^2-k} && Throw@k; k++]; 0]; Array[a, 75] (* Giovanni Resta, Mar 13 2014 *)

Python

import sympy
from sympy import isprime
def c(n):
  for k in range(1,n):
    if isprime(n+k) and isprime(n-k) and isprime(n**2+k) and isprime(n**2-k):
      return k
n = 1
while n < 100:
  if c(n) == None:
    print(0)
  else:
    print(c(n))
  n += 1