A271050 Positive integer k such that k^2 = p^2 + q^2 - 1 where p and q are consecutive primes.

13, 17, 37, 157, 307, 437, 1451, 6487, 60773, 421133, 1445957, 2064493, 15247789, 177075397, 16509255853, 4270704255979, 56635565799013, 124750634536736711, 628179811369719907, 81815870181890275241, 13861061749008806276269, 91566796731172246399571

Offset: 1

Author

Emre APARI, Mar 29 2016

Keywords

• nonn

Comments

Prime terms of this sequence are listed in A167276. - Altug Alkan, Mar 30 2016

Examples

			         7^2 + 11^2 - 1 = 169 (13^2, k is prime),
        11^2 + 13^2 - 1 = 289 (17^2, k is prime),
        23^2 + 29^2 - 1 = 1369 (37^2, k is prime),
      109^2 + 113^2 - 1 = 24649 (157^2, k is prime),
      211^2 + 223^2 - 1 = 94249 (307^2, k is prime),
      307^2 + 311^2 - 1 = 190969 (437^2, k is semiprime),
    1021^2 + 1031^2 - 1 = 2105401 (1451^2, k is prime),
  42967^2 + 42979^2 - 1 = 3693357529 (60773^2, k is prime).
		

Crossrefs

Cf. A001248, A069484, A160054 (the corresponding primes p), A167276.

Programs

Mathematica

p = 2; q = 3; lst = {}; While[p < 10^15, If[ IntegerQ@ Sqrt[p^2 + q^2 - 1], AppendTo[lst, Sqrt[p^2 + q^2 - 1]];
Print[Sqrt[p^2 + q^2 - 1]]]; p = q; q = NextPrime@ q] (* Robert G. Wilson v, Mar 30 2016 *)

PARI

list(nn) = {p = 2; forprime(q=3, nn, if (issquare(s = q^2+p^2-1), print1(sqrtint(s), ", ")); p = q;);} \\ Michel Marcus, Mar 29 2016

Extensions

More terms from Jinyuan Wang, Jan 09 2021