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%I #43 Apr 14 2023 10:54:30
%S 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,
%T 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,
%U 1,6,1,52,5,26,31,26,45,40,11,4,1,20,7,196,19,16
%N a(n) is the least positive integer such that n^2 + a(n)^2 and n^2 + (a(n) - 2)^2 are primes.
%C n + i*a(n) and n + i*(a(n) - 2) are Gaussian twin primes.
%C If n^2 + 1 is a prime then a(n) = 1 else a(n) = A284211(n) + 2.
%H Lars-Erik Svahn, <a href="/A284327/b284327.txt">Table of n, a(n) for n = 1..99</a>
%H Lars-Erik Svahn, <a href="https://github.com/Lehs/ANS-Forth-libraries">numbertheory.4th</a>
%H Akshaa Vatwani, <a href="http://dx.doi.org/10.1016/j.jnt.2016.07.008">Bounded gaps between Gaussian primes</a>, Journal of Number Theory, Volume 171, February 2017, Pages 449-473.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GaussianPrime.html">Gaussian prime</a>
%H <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>
%F a(n) = 1 for n in A005574. - _Michel Marcus_, Mar 31 2017
%e a(1) = 1 since 1^2 + 1^2 = 2 and 1^2 + (1 - 2)^2 = 2 are primes.
%t 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 *)
%o (ANS-Forth)
%o s" numbertheory.4th" included
%o : Gauss_twins \ n -- a(n)
%o dup * locals| square | 0
%o begin 1+ dup dup * square + isprime
%o over 2 - dup * square + isprime and
%o until ;
%o (PARI) a(n) = k=0; while (! (isprime(n^2+k^2) && isprime(n^2+(k-2)^2)), k++); k; \\ _Michel Marcus_, Mar 25 2017
%o (Python)
%o from sympy import isprime
%o def a(n):
%o k=0
%o while True:
%o if isprime(n**2 + k**2) and isprime(n**2 + (k - 2)**2): return k
%o else: k+=1
%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Mar 31 2017
%Y Cf. A005574, A069003, A284211, A284346.
%K nonn
%O 1,3
%A _Lars-Erik Svahn_, Mar 25 2017
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