A069484 -id:A069484 - cp's OEIS frontend

A097431 Integer part of the hypotenuse of right triangles with consecutive prime legs.

3, 5, 8, 13, 17, 21, 25, 29, 37, 42, 48, 55, 59, 63, 70, 79, 84, 90, 97, 101, 107, 114, 121, 131, 140, 144, 148, 152, 157, 169, 182, 189, 195, 203, 212, 217, 226, 233, 240, 248, 254, 263, 271, 275, 280, 290, 307, 318, 322, 326, 333, 339, 347, 359, 367, 376, 381

Offset: 1

Cino Hilliard, Aug 22 2004

nonn

			If legs = 3,5 then floor(sqrt(9+25)) = 5, the 2nd entry.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A069484.

[Floor(Sqrt(NthPrime(n)^2 + NthPrime(n+1)^2)): n in [1..60]]; // Vincenzo Librandi, Mar 11 2015

Table[Floor[Sqrt[Prime[n]^2 + Prime[n + 1]^2]], {n, 60}] (* Vincenzo Librandi, Mar 11 2015 *)
Floor[Sqrt[Total[#^2]]]&/@Partition[Prime[Range[60]],2,1] (* Harvey P. Dale, Mar 30 2024 *)

a(n) = for(j=1,n,x=prime(j);y=prime(j+1);print1(floor(sqrt(x^2+y^2))","))

a(n) = floor(sqrt(prime(n)^2 + prime(n+1)^2)) = floor(sqrt(A069484(n))).

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

Original entry on oeis.org

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

Offset: 1

Emre APARI, Mar 29 2016

nonn

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

			         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).

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

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 *)

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

More terms from Jinyuan Wang, Jan 09 2021

cp's OEIS Frontend

A097431 Integer part of the hypotenuse of right triangles with consecutive prime legs.

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