A165236 Short legs of primitive Pythagorean Triples (a,b,c) such that 2*a+1, 2*b+1 and 2*c+1 are primes.

20, 33, 44, 56, 68, 273, 303, 320, 380, 440, 483, 740, 1071, 1089, 1101, 1220, 1376, 1484, 1635, 1773, 1808, 1869, 1940, 1965, 2000, 2120, 2144, 2204, 2319, 2715, 2763, 3003, 3164, 3309, 3500, 3603, 3729, 3740, 3753, 3801, 4148, 4215, 4323, 4340, 4401

Offset

1,1

Comments

Only one instance of a enters the sequence if multiple solutions exist, like with (a,b,c) = (320,999,1049) and (a,b,c) = (320,25599,25601).

Subsequence of A009004. [R. J. Mathar, Mar 25 2010]

Links

Table of n, a(n) for n=1..45.

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-leg 20 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, a]]]]]]; If[a>amax, Break[]], {m, n+1, 12!, 2}], {n, 1, q, 1}]; Union@lst

Crossrefs

Cf. A009004, A020883, A165237, A165238

Sequence in context: A119873 A364307 A075230 * A067468 A127906 A108667

Adjacent sequences: A165233 A165234 A165235 * A165237 A165238 A165239

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