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%I #5 Apr 16 2024 21:50:11
%S 16,21,24,27,33,36,44,55,56,57,60,63,64,68,75,76,77,81,84,87,88,91,92,
%T 93,96,99,100,104,105,111,115,116,117,119,120,123,124,125,128,129,132,
%U 133,135,136,140,143,144,147,152,153,155,156,160,161,164,165,168,172
%N Short legs in primitive Pythagorean triangles with three side lengths of composite integers.
%C The sequence collects the numbers A such that A^2+B^2 = C^2, A<B<C, gcd(A,B,C) = 1 and such that all three of A, B and C are in A002808. 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.
%e (A,B,C) = (16,63,65) contributes A = 16 to the sequence. (A,B,C) = (21,220,221) contributes A = 21.
%e 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).
%t 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
%Y Cf. A002808, A009004, A020882, A020883, A165158, A165159.
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Sep 06 2009
%E Edited by _R. J. Mathar_, Oct 02 2009