A284327 a(n) is the least positive integer such that n^2 + a(n)^2 and n^2 + (a(n) - 2)^2 are primes.

1, 1, 10, 1, 4, 1, 10, 5, 16, 1, 6, 25, 10, 1, 4, 1, 10, 7, 16, 1, 46, 15, 20, 1, 6, 1, 22, 15, 6, 13, 6, 5, 190, 11, 18, 1, 30, 15, 46, 1, 46, 25, 10, 21, 16, 21, 10, 37, 6, 19, 16, 5, 12, 1, 6, 1, 52, 5, 26, 31, 26, 45, 40, 11, 4, 1, 20, 7, 196, 19, 16

Offset

1,3

Comments

n + i*a(n) and n + i*(a(n) - 2) are Gaussian twin primes.

If n^2 + 1 is a prime then a(n) = 1 else a(n) = A284211(n) + 2.

Links

Lars-Erik Svahn, Table of n, a(n) for n = 1..99

Lars-Erik Svahn, numbertheory.4th

Akshaa Vatwani, Bounded gaps between Gaussian primes, Journal of Number Theory, Volume 171, February 2017, Pages 449-473.

Eric Weisstein's World of Mathematics, Gaussian prime

Index entries for Gaussian integers and primes

Formula

a(n) = 1 for n in A005574. - Michel Marcus, Mar 31 2017

Example

a(1) = 1 since 1^2 + 1^2 = 2 and 1^2 + (1 - 2)^2 = 2 are primes.

Mathematica

Rest@ FoldList[Module[{k = 1}, While[Times @@ Boole@ Map[PrimeQ, {#2^2 + k^2, #2^2 + (k - 2)^2}] < 1, k++]; k] &, 1, Range@ 71] (* Michael De Vlieger, Mar 25 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) = k=0; while (! (isprime(n^2+k^2) && isprime(n^2+(k-2)^2)), k++); k; \\ Michel Marcus, Mar 25 2017
(Python)
from sympy import isprime
def a(n):
    k=0
    while True:
        if isprime(n**2 + k**2) and isprime(n**2 + (k - 2)**2): return k
        else: k+=1
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 31 2017

Crossrefs

Cf. A005574, A069003, A284211, A284346.

Sequence in context: A138261 A068126 A156225 * A322219 A010182 A100925

Adjacent sequences: A284324 A284325 A284326 * A284328 A284329 A284330

Keyword

nonn

Author

Lars-Erik Svahn, Mar 25 2017

Status

approved