cp's OEIS Frontend

A263466 Least k such that prime(n) is the smallest prime p for which k^2 + p^2 is also prime, or 0 if none.

%I A263466 #18 Nov 07 2015 17:03:38
%S A263466 1,2,4,12,14,48,178,44,152,66,224,272,496,322,408,2068,114,354,592,
%T A263466 584,3192,406,2708,774,2658,394,4102,2432,3346,2562,8722,4424,9562,
%U A263466 2986,6856,1714,21318,5858,7568,16272,7576,4864,6244,29262,29992,9996,10406,58348,16872,11384,12738,22126,9946,24214,81682,46082,74616,88016,6788,30856,21542,38672,131492,62874,75358,95262,39554,83552,65022,73664
%N A263466 Least k such that prime(n) is the smallest prime p for which k^2 + p^2 is also prime, or 0 if none.
%C A263466 The data support the conjecture in A263977 that if k > 0 is even, then k^2 + p^2 is prime for some prime p.
%C A263466 a(n) is the location of the first occurrence of prime(n) in A263978.
%H A263466 Jonathan Sondow and Robert G. Wilson v, <a href="/A263466/b263466.txt">Table of n, a(n) for n = 1..260</a>
%H A263466 Stephan Baier and Liangyi Zhao, <a href="http://arxiv.org/abs/math/0703284">On Primes Represented by Quadratic Polynomials</a>, Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.
%H A263466 Étienne Fouvry and Henryk Iwaniec, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa79/aa7935.pdf">Gaussian primes</a>, Acta Arithmetica 79:3 (1997), pp. 249-287.
%e A263466 The primes p < prime(3) = 5 are p = 2 and 3. As 1^2 + 2^2 = 5, 2^2 + 3^2 = 13, and 3^2 + 2^2 = 13 are prime, a(3) >= 4. But 4^2 + 2^2 = 20 and 4^2 + 3^2 = 25 are not prime, while 4^2 + 5^2 = 41 is prime, so a(3) = 4.
%t A263466 f[n_] := Block[{p = 2}, While[ !PrimeQ[n^2 + p^2], p = NextPrime@ p]; p]; t = 0*Range@ 300; t[[1]] = 1; k = 2; While[k < 50000001, p = f@ k; If[ t[[PrimePi@ p]] == 0, t[[PrimePi@ p]] = k; Print[{PrimePi@ p, p, k}]]; k += 2]; t
%Y A263466 Cf. A240130, A240131, A263721, A263722, A263726, A263977, A263978.
%K A263466 nonn
%O A263466 1,2
%A A263466 _Jonathan Sondow_ and _Robert G. Wilson v_, Nov 02 2015