A165238 Hypotenuses c of primitive Pythagorean Triples (a,b,c) such that 2*a+1, 2*b+1 and 2*c+1 are primes.

29, 65, 293, 485, 785, 1049, 1469, 1961, 2105, 3005, 3725, 3821, 4145, 4181, 4685, 4745, 5105, 5501, 6053, 6929, 6953, 7121, 7361, 7841, 8693, 9029, 9125, 10025, 12041, 12833, 12965, 13649, 14285, 14909, 15173, 15689, 15773, 15821, 16493

Offset

1,1

Comments

Only one copy of c enters the sequence if multiple solutions exist, like with (a,b,c) = (3164,12573,12965) and (a,b,c) = (483,12956,12965).

Subsequence of A020882. [R. J. Mathar, Mar 25 2010]

Links

Table of n, a(n) for n=1..39.

Eric Weisstein, Pythagorean Triple, MathWorld

Example

(a,b,c) = (20,21,29), (33,56,65), (44,483,485), (56,783,785), (68,285,293), (273,4136,4145).
In the first case, for example, 2*20+1=41, 2*21+1 and 2*29+1 are all prime, which adds the half-hypotenuse 29 to the sequence.

Mathematica

amax=6*10^4; lst={}; k=0; q=12!; Do[If[(e=((n+1)^2-n^2))>amax, Break[]];
Do[If[GCD[m, n]==1, a=m^2-n^2; If[PrimeQ[2*a+1], b=2*m*n; If[PrimeQ[2*b+1],
If[GCD[a, b]==1, If[a>b, {a, b}={b, a}]; If[a>amax, Break[]]; c=m^2+n^2;
If[PrimeQ[2*c+1], k++; AppendTo[lst, c]]]]]]; If[a>amax, Break[]], {m, n+1, 12!, 2}], {n, 1, q, 1}]; Union@lst

Crossrefs

Cf. A020882, A020883, A165236, A165237

Sequence in context: A044131 A044512 A211492 * A201022 A367151 A118481

Adjacent sequences: A165235 A165236 A165237 * A165239 A165240 A165241

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Sep 09 2009

Extensions

Comments moved to examples and definition clarified by R. J. Mathar, Mar 25 2010

Status

approved