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

2,1

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.

References

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

Links

Table of n, a(n) for n=2..90.

Wikipedia, Schinzel's Hypothesis H

Example

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.

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}]

Crossrefs

Cf. A263724.

Sequence in context: A053009 A377465 A069485 * A340053 A174056 A307894

Adjacent sequences: A263722 A263723 A263724 * A263726 A263727 A263728

Keyword

nonn

Author

Jonathan Sondow, Oct 24 2015

Status

approved