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
Keywords
A165160 Short legs in primitive Pythagorean triangles with three side lengths of composite integers.
16, 21, 24, 27, 33, 36, 44, 55, 56, 57, 60, 63, 64, 68, 75, 76, 77, 81, 84, 87, 88, 91, 92, 93, 96, 99, 100, 104, 105, 111, 115, 116, 117, 119, 120, 123, 124, 125, 128, 129, 132, 133, 135, 136, 140, 143, 144, 147, 152, 153, 155, 156, 160, 161, 164, 165, 168, 172
Offset: 1
Keywords
Comments
The sequence collects the numbers A such that A^2+B^2 = C^2, AA002808. If there are two or more triangles of this kind with the same A, like (A,B,C) = (33,544,545) and (A,B,C) = (33,56,65), only one instance of A is added to the sequence.
Examples
(A,B,C) = (16,63,65) contributes A = 16 to the sequence. (A,B,C) = (21,220,221) contributes A = 21. Further length triples are (24,143,145), (27,364,365), (33,56,65), (33,544,545), (36,77,85), (36,323,325), (44,117,125), (44,483,485), (55,1512,1513), (56,783,785), (57,176,185).
Programs
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Mathematica
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,a]]],{b,a+1,Floor[a^2/2],1}], {a,3,400,1}]; Union@lst
Extensions
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.
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
Examples
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; ...
Crossrefs
Programs
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Mathematica
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
Comments
Examples
Crossrefs
Programs
Mathematica
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@lstExtensions