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

%I A271050 #44 Jan 09 2021 07:31:02
%S A271050 13,17,37,157,307,437,1451,6487,60773,421133,1445957,2064493,15247789,
%T A271050 177075397,16509255853,4270704255979,56635565799013,
%U A271050 124750634536736711,628179811369719907,81815870181890275241,13861061749008806276269,91566796731172246399571
%N A271050 Positive integer k such that k^2 = p^2 + q^2 - 1 where p and q are consecutive primes.
%C A271050 Prime terms of this sequence are listed in A167276. - _Altug Alkan_, Mar 30 2016
%e A271050          7^2 + 11^2 - 1 = 169 (13^2, k is prime),
%e A271050         11^2 + 13^2 - 1 = 289 (17^2, k is prime),
%e A271050         23^2 + 29^2 - 1 = 1369 (37^2, k is prime),
%e A271050       109^2 + 113^2 - 1 = 24649 (157^2, k is prime),
%e A271050       211^2 + 223^2 - 1 = 94249 (307^2, k is prime),
%e A271050       307^2 + 311^2 - 1 = 190969 (437^2, k is semiprime),
%e A271050     1021^2 + 1031^2 - 1 = 2105401 (1451^2, k is prime),
%e A271050   42967^2 + 42979^2 - 1 = 3693357529 (60773^2, k is prime).
%t A271050 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]];
%t A271050 Print[Sqrt[p^2 + q^2 - 1]]]; p = q; q = NextPrime@ q] (* _Robert G. Wilson v_, Mar 30 2016 *)
%o A271050 (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
%Y A271050 Cf. A001248, A069484, A160054 (the corresponding primes p), A167276.
%K A271050 nonn
%O A271050 1,1
%A A271050 _Emre APARI_, Mar 29 2016
%E A271050 More terms from _Jinyuan Wang_, Jan 09 2021