A165158 Hypotenuses of primitive Pythagorean triangles such that all 3 sides are composite.

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

Author

Vladimir Joseph Stephan Orlovsky, Sep 06 2009

Keywords

• nonn

Comments

Numbers C in triples of the form A^2+B^2=C^2, gcd(A,B,C)=1 and all of A, B and C in A002808.

If multiple solutions exist for the same C, as for example (A,B,C) = (16,63,65) and (33,56,65),

only one instance of C is added to the sequence.

Examples

			(A,B,C) = (16,63,65), (36,77,85), (44,117,125) etc
		

Crossrefs

Cf. A020882, A165159

Programs

Mathematica

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 *)

Extensions

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