A165158 - OEIS

%I #10 Aug 27 2024 15:26:18

%S 65,85,125,145,169,185,205,221,265,289,305,325,365,377,425,445,481,

%T 485,493,505,533,545,565,625,629,685,689,697,725,745,785,793,845,865,

%U 901,905,925,949,965,985,1025,1037,1073,1105,1145,1157,1165,1189,1205,1241

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

%C 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.

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

%C only one instance of C is added to the sequence.

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

%t 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

%t 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 *)

%Y Cf. A020882, A165159

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Sep 06 2009

%E Typo in description corrected by _Alan Frank_, Oct 09 2009

%E Definition clarified, comment moved to the examples and new comment added - _R. J. Mathar_, Oct 21 2009