A165158
Hypotenuses of primitive Pythagorean triangles such that all 3 sides are composite.
Original entry on oeis.org
65, 85, 125, 145, 169, 185, 205, 221, 265, 289, 305, 325, 365, 377, 425, 445, 481, 485, 493, 505, 533, 545, 565, 625, 629, 685, 689, 697, 725, 745, 785, 793, 845, 865, 901, 905, 925, 949, 965, 985, 1025, 1037, 1073, 1105, 1145, 1157, 1165, 1189, 1205, 1241
Offset: 1
(A,B,C) = (16,63,65), (36,77,85), (44,117,125) etc
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lst={};Do[Do[If[IntegerQ[a=Sqrt[c^2-b^2]]&&GCD[a,b,c]==1,If[a>=b,Break[]]; If[ !PrimeQ[a]&&!PrimeQ[b]&&!PrimeQ[c],AppendTo[lst,c]]],{b,c-1,4, -1}],{c,5,2000,1}];Union@lst
Select[Sort[{Numerator[#],Denominator[#],Sqrt[Numerator[#]^2+Denominator[#]^2]}&/@ Union[ #[[1]]/#[[2]]&/@Union[Sort/@Select[Select[Flatten[Outer[List,Range[1500],Range[ 1500]],1],#[[1]]!=#[[2]]&],IntegerQ[Sqrt[#[[1]]^2+#[[2]]^2]]&]]]],AllTrue[#,CompositeQ]&][[;;,3]]//Union (* Harvey P. Dale, Aug 27 2024 *)
Typo in description corrected by
Alan Frank, Oct 09 2009
Definition clarified, comment moved to the examples and new comment added -
R. J. Mathar, Oct 21 2009
A165160
Short legs in primitive Pythagorean triangles with three side lengths of composite integers.
Original entry on oeis.org
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
(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).
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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
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.
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