A243145 Least positive number k such that n+k and n+k^2 are both prime.

1, 1, 2, 1, 6, 1, 4, 3, 2, 1, 6, 1, 4, 3, 2, 1, 6, 1, 12, 3, 16, 1, 6, 7, 4, 15, 2, 1, 12, 1, 6, 9, 8, 3, 6, 1, 4, 3, 2, 1, 30, 1, 4, 3, 8, 1, 6, 5, 10, 3, 10, 1, 6, 5, 4, 15, 2, 1, 42, 1, 6, 21, 4, 3, 6, 1, 4, 15, 2, 1, 30, 1, 6, 33, 8, 25, 6, 1, 10, 3, 16, 1, 24, 5, 4, 15

Offset

1,3

Comments

For n > 1, a(n) == n+1 (mod 2). a(n) = 1 for n in A006093. - Robert Israel, Feb 02 2018

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

8+1 and 8+1^2 (9) isn't prime. 8+2 and 8+2^2 (10 and 12) aren't both prime. But 8+3 and 8+3^2 (11 and 17) are both prime. Thus a(8) = 3.

Maple

f:= proc(n) local k;
  for k from (n mod 2)+1 by 2 do
    if isprime(n+k) and isprime(n+k^2) then return k fi
  od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Feb 02 2018

Mathematica

f[n_] := Module[{k}, For[k = Mod[n, 2] + 1, True, k += 2, If[PrimeQ[n + k] && PrimeQ[n + k^2], Return[k]]]]; f[1] = 1; f /@ Range[100] (* Jean-François Alcover, Feb 03 2018, after Robert Israel *)

Prog

(PARI) a(n)=for(k=1, 10^6, if(ispseudoprime(n+k)&&ispseudoprime(n+k^2), return(k)))
n=1; while(n<100, print1(a(n), ", "); n+=1)
(Python)
from sympy import isprime, nextprime
def A243145(n):
    m = n
    while True:
        m = nextprime(m)
        k = m-n
        if isprime(n+k**2):
            return k #  Chai Wah Wu, Sep 03 2019

Crossrefs

Cf. A006093.

Sequence in context: A321725 A154744 A285038 * A306695 A242926 A189733

Adjacent sequences: A243142 A243143 A243144 * A243146 A243147 A243148

Keyword

nonn

Author

Derek Orr, May 30 2014

Status

approved