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%I #29 May 19 2023 07:45:53
%S 65,85,145,185,205,221,265,305,325,365,377,425,445,481,485,493,505,
%T 533,545,565,629,685,689,697,725,745,785,793,845,865,901,905,925,949,
%U 965,985,1025,1037,1073,1105,1145,1157,1165,1189,1205,1241,1261,1285,1313,1325
%N Hypotenuses of more than one primitive Pythagorean triangle.
%C The subsequence allowing 4 different shapes is in A159781. [_R. J. Mathar_, Apr 12 2010]
%C A024362(a(n)) > 1. - _Reinhard Zumkeller_, Dec 02 2012
%H Ray Chandler, <a href="/A024409/b024409.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Reinhard Zumkeller)
%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 65^2 = 16^2 + 63^2 = 33^2 + 56^2 (also = 25^2 + 60^2 = 39^2 + 52^2, but these are not primitive, with gcd = 5 resp. 13). Note that 65 = 1^2 + 8^2 = 4^2 + 7^2 is also the least integer > 1 to be a sum a^2 + b^2 with gcd(a,b) = 1 in two ways. - _M. F. Hasler_, May 18 2023
%t f[c_] := f[c] = Block[{a = 1, b, cnt = 0, lmt = Floor[ Sqrt[c^2/2]]}, While[b = Sqrt[c^2 - a^2]; a < lmt, If[IntegerQ@ b && GCD[a, b, c] == 1, cnt++]; a++]; cnt]Select[1 + 4 Range@ 335, f@# > 1 &] (* _Robert G. Wilson v_, Mar 16 2014 *)
%t Select[Tally[Sqrt[Total[#^2]]&/@Union[Sort/@({Times@@#,(Last[#]^2-First[ #]^2)/2}&/@(Select[Subsets[Range[1,71,2],{2}],GCD@@# == 1&]))]],#[[2]]> 1&][[All,1]]//Sort (* _Harvey P. Dale_, Sep 29 2018 *)
%o (Haskell)
%o import Data.List (findIndices)
%o a024409 n = a024409_list !! (n-1)
%o a024409_list = map (+ 1) $ findIndices (> 1) a024362_list
%o -- _Reinhard Zumkeller_, Dec 02 2012
%Y Cf. A020882, A120960, subsequence of A008846.
%K nonn
%O 1,1
%A _David W. Wilson_
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