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A263725 Smallest prime q > prime(n+3) such that the number p = prime(n)^2 + prime(n+1)^2 + prime(n+2)^2 + prime(n+3)^2 + q^2 is also prime.

%I A263725 #7 Oct 31 2015 14:07:36
%S A263725 13,17,37,31,31,37,41,41,43,47,59,61,89,79,71,79,79,89,97,109,127,107,
%T A263725 109,109,113,139,131,139,151,149,157,157,173,181,173,191,191,193,197,
%U A263725 223,199,211,233,239,229,233,263,257,263,271,271,277,271,281,281,293,293,311,349,317,353,331,353,353,359,419,359,419,379,419,397,401,431,409,409,433,461,443,487,449,541,487,463,569,479,467,487,491,503
%N A263725 Smallest prime q > prime(n+3) such that the number p = prime(n)^2 + prime(n+1)^2 + prime(n+2)^2 + prime(n+3)^2 + q^2 is also prime.
%C A263725 The corresponding primes p form A263724.
%C A263725 The prime q exists for all n > 1 under Schinzel's Hypothesis H; see Sierpinski (1988), p. 221.
%D A263725 W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988.
%H A263725 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's Hypothesis H</a>
%e A263725 The primes 373 = 3^2 + 5^2 + 7^2 + 11^2 + 13^2, 653 = 5^2 + 7^2 + 11^2 + 13^2 + 17^2, and 1997 = 7^2 + 11^2 + 13^2 + 17^2 + 37^2 lead to a(2) = 13, a(3) = 17, and a(4) = 37.
%t A263725 Table[k = 4;
%t A263725  While[! PrimeQ[Sum[Prime[n + j]^2, {j, 0, 3}] + Prime[n + k]^2], k++];
%t A263725  Prime[n + k], {n, 2, 90}]
%Y A263725 Cf. A263724.
%K A263725 nonn
%O A263725 2,1
%A A263725 _Jonathan Sondow_, Oct 24 2015