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A067755 Even legs of Pythagorean triangles whose other leg and hypotenuse are both prime.

%I A067755 #18 Apr 23 2023 23:42:12
%S A067755 4,12,60,180,420,1740,1860,2520,3120,5100,8580,9660,16380,19800,36720,
%T A067755 60900,71820,83640,100800,106260,135720,161880,163020,199080,205440,
%U A067755 218460,273060,282000,337020,388080,431520,491040,531480,539760,552300
%N A067755 Even legs of Pythagorean triangles whose other leg and hypotenuse are both prime.
%C A067755 Apart from the first two terms, every term is divisible by 60 and is of the form 450*k^2 +/- 30*k or 450*k^2 +/- 330*k + 60 for some k.
%C A067755 In such a triangle, this even leg is always the longer leg, and the hypotenuse = a(n) + 1. The Pythagorean triples are (A048161(n), a(n), A067756(n)), so, for a(2) = 12, the corresponding Pythagorean triple is (5, 12, 13). - _Bernard Schott_, Apr 12 2023
%H A067755 Ray Chandler, <a href="/A067755/b067755.txt">Table of n, a(n) for n = 1..10000</a>
%H A067755 H. Dubner and T. Forbes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/DUBNER/pyth.html">Prime Pythagorean triangles</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.3.
%F A067755 a(n) = (A048161(n)^2 - 1)/2 = A067756(n) - 1.
%e A067755 4 is a term: in the right triangle (3, 4, 5), 3 and 5 are prime.
%e A067755 5100 is a term: in the right triangle (101, 5100, 5101), 101 and 5101 are prime.
%t A067755 lst={}; Do[q=(Prime[n]^2+1)/2; If[PrimeQ[q], AppendTo[lst, (Prime[n]^2-1)/2]], {n, 200}]; lst (* _Frank M Jackson_, Nov 02 2013 *)
%Y A067755 Cf. A048161, A067756.  Contains every value of A051858.
%K A067755 nonn
%O A067755 1,1
%A A067755 _Henry Bottomley_, Jan 31 2002