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A284346 a(n) is the least positive integer such that n^2 + a(n)^2 and (n + 1)^2 + (a(n) + 1)^2 are primes. 3

%I #43 Feb 28 2023 11:22:35

%S 2,1,8,1,4,1,2,3,16,3,6,7,8,1,4,1,22,5,6,3,4,17,18,5,4,1,32,5,10,29,4,

%T 27,8,15,18,1,2,15,10,3,4,247,8,15,14,19,22,35,6,19,4,27,10,11,8,1,2,

%U 5,40,13,44,127,58,61,28,1,22,13,10,19,6,7,8,15,4,9

%N a(n) is the least positive integer such that n^2 + a(n)^2 and (n + 1)^2 + (a(n) + 1)^2 are primes.

%C n is odd iff a(n) is even.

%H Lars-Erik Svahn, <a href="/A284346/b284346.txt">Table of n, a(n) for n = 1..10000</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>.

%e a(1)=2 since (1 + 1)^2 + (1 + 1)^2 is not prime, but 1^2 + 2^2 = 5 and (1 + 1)^2 + (2 + 1)^2 = 13 are prime.

%t Rest@ FoldList[Module[{k = 1}, While[Times @@ Boole@ Map[PrimeQ, {#2^2 + k^2, (#2 + 1)^2 + (k + 1)^2}] < 1, k++]; k] &, 1, Range@ 76] (* _Michael De Vlieger_, Mar 25 2017 *)

%o (ANS-Forth)

%o s" numbertheory.4th" included

%o : Gauss_twin \ n -- a(n)

%o locals| n | 0

%o begin 1+ dup dup * n dup * + isprime

%o over 1+ dup * n 1+ dup * + isprime and

%o until ;

%o (PARI) a(n) = my(k=0); while (! (isprime(n^2+k^2) && isprime((n+1)^2+(k+1)^2)), k++); k; \\ _Michel Marcus_, Mar 25 2017

%Y Cf. A069003, A284211, A284327.

%K nonn

%O 1,1

%A _Lars-Erik Svahn_, Mar 25 2017

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)