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

2,1

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).

References

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

Links

Table of n, a(n) for n=2..60.

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

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}]

Crossrefs

Cf. A133559, A263725.

Sequence in context: A142395 A182573 A142921 * A133559 A229499 A023313

Adjacent sequences: A263721 A263722 A263723 * A263725 A263726 A263727

Keyword

nonn

Author

Jonathan Sondow, Oct 24 2015

Status

approved