A165160 -id:A165160 - cp's OEIS frontend

A165159 Long legs in primitive Pythagorean triangles with three side lengths of composite integers.

56, 63, 77, 117, 120, 143, 153, 156, 171, 176, 187, 220, 224, 240, 247, 253, 273, 304, 323, 345, 352, 357, 360, 364, 377, 396, 403, 416, 435, 437, 456, 460, 468, 475, 476, 483, 493, 513, 525, 527, 528, 544, 561, 621, 624, 627, 644, 663, 665, 667, 672, 680

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 06 2009

nonn

The sequence collects the numbers B such that A^2+B^2=C^2, A

three of A, B and C are in A002808. If there are two or more triangles of this kind with the same B,

like (A,B,C) = (1003,1596,1885) and (A,B,C) = (1403,1596,2125), only one instance

of B is added to the sequence.

			(A,B,C)=(33,56,65) contributes B=56 to the sequence. (A,B,C)=(16,63,65) contributes B=63 to the sequence.

Cf. A020882, A020883, A165158, A165160.

lst={}; Do[Do[If[IntegerQ[c=Sqrt[a^2+b^2]] && GCD[a,b,c]==1,If[ !PrimeQ[a]&&!PrimeQ[b] && !PrimeQ[c], AppendTo[lst,b]]],{a,b-1,3,-1}], {b,4,2000,1}];Union@lst

Edited by R. J. Mathar, Oct 02 2009

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

Vladimir Joseph Stephan Orlovsky, Sep 11 2009

			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.

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

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A165159 Long legs in primitive Pythagorean triangles with three side lengths of composite integers.

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