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

1,1

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.

Links

Table of n, a(n) for n=1..50.

Example

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

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 *)

Crossrefs

Cf. A020882, A165159

Sequence in context: A024508 A025303 A071011 * A084648 A224770 A274044

Adjacent sequences: A165155 A165156 A165157 * A165159 A165160 A165161

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Sep 06 2009

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

Status

approved