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

1,1

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.

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.

Example

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.

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 *)

Prog

(ANS-Forth)
s" numbertheory.4th" included
: Gauss_twins \ n -- a(n)
  dup * locals| square | 0
  begin 1+ dup dup * square + isprime
     over 2 + dup * square + isprime and
  until ;
(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

Crossrefs

Cf. A069003.

Sequence in context: A036296 A078105 A075513 * A246403 A355183 A258502

Adjacent sequences: A284208 A284209 A284210 * A284212 A284213 A284214

Keyword

nonn

Author

Lars-Erik Svahn, Mar 23 2017

Status

approved