A165236
Short legs of primitive Pythagorean Triples (a,b,c) such that 2*a+1, 2*b+1 and 2*c+1 are primes.
Original entry on oeis.org
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
(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.
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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
Comments moved to examples and definition clarified by
R. J. Mathar, Mar 25 2010
A165238
Hypotenuses c of primitive Pythagorean Triples (a,b,c) such that 2*a+1, 2*b+1 and 2*c+1 are primes.
Original entry on oeis.org
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
(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.
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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
Comments moved to examples and definition clarified by
R. J. Mathar, Mar 25 2010
A165262
Sorted hypotenuses with no repeats of Primitive Pythagorean Triples (PPT) if sum of all 3 sides are averages of twin prime pairs.
Original entry on oeis.org
5, 13, 85, 113, 145, 197, 221, 241, 349, 457, 541, 569, 625, 821, 829, 841, 1025, 1037, 1093, 1157, 1241, 1433, 1465, 1621, 1741, 1769, 2029, 2069, 2249, 2353, 2441, 2465, 2501, 2669, 2725, 2801, 2809, 2825, 2873, 3029, 3077, 3221, 3293, 3305, 3389, 3889
Offset: 1
Triples begin 3,4,5; 5,12,13; 15,112,113; 21,220,221; 24,143,145; 28,195,197; 36,77,85; 41,840,841; 59,1740,1741; 64,1023,1025; 89,3960,3961; 100,2499,2501; ...
So with sorted hypotenuses:
3 + 4 + 5 = 12, and 11 and 13 are twin primes;
5 + 12 + 13 = 30, and 29 and 31 are twin primes; ...
Cf.
A009004,
A020882,
A020883,
A165158,
A165159,
A165160,
A165236,
A165237,
A165238,
A165260,
A165261.
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amax=10^5; 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; b=2*m*n; If[GCD[a,b]==1,If[a>b,{a,b}={b,a}]; If[a>amax,Break[]]; c=m^2+n^2; x=a+b+c; If[PrimeQ[x-1]&&PrimeQ[x+1],k++; AppendTo[lst,c]]]],{m,n+1,12!,2}],{n,1,q,1}]; Union@lst
Showing 1-3 of 3 results.
Comments