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

1,8

Comments

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A082467, A172990, A172989.

Sequence in context: A134676 A286246 A286249 * A324052 A103491 A268932

Adjacent sequences: A239143 A239144 A239145 * A239147 A239148 A239149

Keyword

nonn

Author

Derek Orr, Mar 11 2014

Status

approved