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

Author

Lars-Erik Svahn, Mar 25 2017

Keywords

• nonn

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.

Examples

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

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

Crossrefs

Cf. A005574, A069003, A284211, A284346.

Programs

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 *)

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

Formula

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