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

Author

Vladimir Joseph Stephan Orlovsky, Sep 09 2009

Keywords

• nonn

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]

Examples

			(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.
		

Links

• Eric Weisstein, Pythagorean Triple, MathWorld

Crossrefs

Cf. A020882, A020883, A165236, A165237

Programs

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

Extensions

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