A165236 - OEIS

%I #3 Mar 31 2012 12:38:26

%S 20,33,44,56,68,273,303,320,380,440,483,740,1071,1089,1101,1220,1376,

%T 1484,1635,1773,1808,1869,1940,1965,2000,2120,2144,2204,2319,2715,

%U 2763,3003,3164,3309,3500,3603,3729,3740,3753,3801,4148,4215,4323,4340,4401

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

%C 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).

%C Subsequence of A009004. [_R. J. Mathar_, Mar 25 2010]

%e (a,b,c) = (20,21,29), (33,56,65), (44,483,485), (56,783,785), (68,285,293), (273,4136,4145).

%e 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.

%t amax=6*10^4;lst={};k=0;q=12!;Do[If[(e=((n+1)^2-n^2))>amax,Break[]];

%t 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[]];

%t 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

%Y Cf. A009004, A020883, A165237, A165238

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Sep 09 2009

%E Comments moved to examples and definition clarified by _R. J. Mathar_, Mar 25 2010