A165261
Long legs of primitive Pythagorean triples which have a perimeter which is the average of a twin prime pair.
Original entry on oeis.org
4, 12, 77, 112, 143, 195, 209, 220, 299, 420, 425, 520, 527, 629, 700, 840, 868, 988, 1023, 1085, 1127, 1209, 1305, 1421, 1480, 1720, 1740, 1900, 2001, 2021, 2255, 2296, 2320, 2331, 2332, 2499, 2520, 2548, 2583, 2604, 2752, 2829, 2964, 3021, 3256, 3311
Offset: 1
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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,b]]]],{m,n+1,12!,2}],{n,1,q,1}];Union@lst
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 1-2 of 2 results.