A284211 a(n) is the least positive integer such that n^2 + a(n)^2 and n^2 + (a(n) + 2)^2 are primes.

2, 1, 8, 9, 2, 29, 8, 3, 14, 1, 4, 23, 8, 9, 2, 29, 8, 5, 14, 1, 44, 13, 18, 59, 4, 9, 20, 13, 4, 11, 4, 3, 188, 9, 16, 149, 28, 13, 44, 1, 44, 23, 8, 19, 14, 19, 8, 35, 4, 17, 14, 3, 10, 59, 4, 9, 50, 3, 24, 29, 24, 43, 38, 9, 2, 89, 18, 5, 194, 17, 14, 5

Offset: 1

Author

Lars-Erik Svahn, Mar 23 2017

Keywords

• nonn

Comments

z = n + i*a(n) and z' = n + i*(a(n) + 2) are two Gaussian primes such that |z - z'| = 2, corresponding to twin primes.

Examples

			a(1)=2: 1^2 + 1^2 = 2 is a prime but 1 + (1 + 2)^2 = 10 is not, while 1^2 + 2^2 = 5 and 1^2 + (2+2)^2 = 17 are both primes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (first 99 terms from Lars-Erik Svahn)
• Lars-Erik Svahn, numbertheory.4th
• Akshaa Vatwani, Bounded gaps between Gaussian primes, J. of Number Theory 171 (2017), 449-473.

Crossrefs

Cf. A069003.

Programs

Maple

f:= proc(n) local k,pp,p;
    pp:= false;
    for k from (n+1) mod 2 by 2 do
      p:= isprime(n^2 + k^2);
      if p and pp then return k-2 fi;
      pp:= p;
    od;
end proc:
map(f, [$1..100]); # Robert Israel, Mar 30 2017

Mathematica

a[n_] := Block[{k = Mod[n, 2] + 1}, While[! PrimeQ[n^2 + k^2] || ! PrimeQ[n^2 + (k + 2)^2], k += 2]; k]; Array[a, 72] (* Giovanni Resta, Mar 23 2017 *)
lpi[n_]:=Module[{k=1},While[!AllTrue[n^2+{k^2,(k+2)^2},PrimeQ],k++];k]; Array[lpi,80] (* Harvey P. Dale, Aug 15 2024 *)

PARI

a(n) = my(k=n%2+1); while (!(isprime(n^2+k^2) && isprime(n^2+(k+2)^2)), k+=2); k  \\ Michel Marcus, Mar 25 2017