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

1,1

Comments

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

Links

Table of n, a(n) for n=1..22.

Example

         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).

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A246944 A086402 A358571 * A039290 A045267 A043113

Adjacent sequences: A271047 A271048 A271049 * A271051 A271052 A271053

Keyword

nonn

Author

Emre APARI, Mar 29 2016

Extensions

More terms from Jinyuan Wang, Jan 09 2021

Status

approved