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.

13, 17, 37, 31, 31, 37, 41, 41, 43, 47, 59, 61, 89, 79, 71, 79, 79, 89, 97, 109, 127, 107, 109, 109, 113, 139, 131, 139, 151, 149, 157, 157, 173, 181, 173, 191, 191, 193, 197, 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

Offset: 2

Author

Jonathan Sondow, Oct 24 2015

Keywords

• nonn

Comments

The corresponding primes p form A263724.

The prime q exists for all n > 1 under Schinzel's Hypothesis H; see Sierpinski (1988), p. 221.

Examples

			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.
		

References

• W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988.

Links

• Wikipedia, Schinzel's Hypothesis H

Crossrefs

Cf. A263724.

Programs

Mathematica

Table[k = 4;
 While[! PrimeQ[Sum[Prime[n + j]^2, {j, 0, 3}] + Prime[n + k]^2], k++];
 Prime[n + k], {n, 2, 90}]