|
%I #23 Oct 22 2021 23:48:59
%S 420,572,780,840,924,1020,1292,1596,1680,1716,1848,1932,2100,2145,
%T 2244,2300,2484,2520,2640,2652,2700,2900,2964,3080,3132,3315,3348,
%U 3432,3465,3596,3640,3828,3876,3960,4060,4092,4095,4340,4488,4588,4620,4680,4692
%N Long leg of more than one primitive Pythagorean triangle.
%C Also, middle side a of more than one primitive integer-sided triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c; hence, terms that appear more than once in A020883. - _Bernard Schott_, Oct 21 2021
%H Ray Chandler, <a href="/A024410/b024410.txt">Table of n, a(n) for n = 1..10000</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%e From _Bernard Schott_, Oct 21 2021: (Start)
%e -> For primitive Pythagorean triples:
%e a(1) = 420 because 420 is the smallest long leg that belongs to more than one primitive Pythagorean triples, we have 29^2 + 420^2 = 421^2 and 341^2 + 420^2 = 541^2.
%e -> For primitive triples with 2/a = 1/b + 1/c:
%e a(1) = 420 because 420 is the smallest middle side a that belongs to more than one primitive integer-sided triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c, we have 2/420 = 1/310 + 1/651 and 2/420 = 1/406 + 1/435. (End)
%t bb=1;s=e="";For[b=1,b<=12^3,For[a=b-1,a>2,c=(a^2+b^2)^0.5;If[c==Round[c]&&GCD[a,b]==1,If[b==bb,e=e<>ToString[b]<>",";s=s<>ToString[a]<>","<>ToString[b]<>","<>ToString[Round[c]]<>"; "];bb=b];a-- ];b++ ];Print["B = ",e] (* _Vladimir Joseph Stephan Orlovsky_, Apr 29 2008 *)
%t Select[Tally[Union[Sort/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@(Select[ Subsets[Range[1,121,2],{2}],GCD@@#==1&]))][[All,2]]],#[[2]]>1&][[All,1]] //Sort (* _Harvey P. Dale_, Mar 07 2020 *)
%Y Cf. A020883.
%K nonn
%O 1,1
%A _David W. Wilson_
|
