A263724 Least prime p = prime(n)^2 + prime(n+1)^2 + prime(n+2)^2 + prime(n+3)^2 + q^2, where q > prime(n+3) is also prime.

373, 653, 1997, 1901, 2309, 3389, 4373, 5381, 6701, 8069, 10589, 12269, 18269, 18461, 19541, 22973, 24821, 29021, 32909, 38261, 46589, 45869, 50021, 53549, 56909, 66029, 69389, 77261, 87629, 93581, 102101, 107741, 118901, 128981, 131837, 145517, 152909, 159869, 170021, 188261, 184901, 196661, 214469, 229781, 237821, 252509, 277157, 281429, 291101, 305933, 317693, 333029, 344021, 359981, 370661, 387341, 395069, 418349, 460949

Offset: 2

Author

Jonathan Sondow, Oct 24 2015

Keywords

• nonn

Comments

The corresponding prime q is in A263725.

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

If q = prime(n+4), then p is in A133559 (prime sums of squares of 5 consecutive primes). The converse holds if a(n) != a(m) when n != m (which holds if a(n) < a(n+1), as appears to be true).

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(1) = 373, a(2) = 653, and a(3) = 1997.
		

References

• W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988; see p. 221.

Links

• Wikipedia, Schinzel's Hypothesis H

Crossrefs

Cf. A133559, A263725.

Programs

Mathematica

Table[k = 4;
While[p = Sum[Prime[n + j]^2, {j, 0, 3}] + Prime[n + k]^2; ! PrimeQ[p],
  k++]; p, {n, 2, 60}]