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%I #45 Jun 13 2022 10:03:01
%S 6,30,60,84,180,210,330,504,546,630,840,924,990,1224,1320,1386,1560,
%T 1710,1716,2310,2340,2574,2730,3036,3570,3900,4080,4290,4620,4914,
%U 5016,5610,5814,6090,6630,7140,7440,7854,7956,7980,8970,8976,9690,10374
%N Areas of right triangles with coprime integer sides.
%C Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives areas A*B/2.
%C By Theorem 2 of Mohanty and Mohanty, all these numbers are primitive Pythagorean. - _T. D. Noe_, Sep 24 2013
%C This sequence also gives Fibonacci's congruous numbers (without multiplicity, in increasing order) divided by 4. See A258150. - _Wolfdieter Lang_, Jun 14 2015
%C The same as A024406 with duplicates removed. All terms are multiples of 6, cf. A258151. - _M. F. Hasler_, Jan 20 2019
%H T. D. Noe, <a href="/A024365/b024365.txt">Table of n, a(n) for n = 1..10000</a> (corrected by _Giovanni Resta_, Jan 21 2019)
%H Supriya Mohanty and S. P. Mohanty, <a href="http://www.fq.math.ca/Scanned/28-1/mohanty.pdf">Pythagorean Numbers</a>, Fibonacci Quarterly 28 (1990), 31-42.
%F Positive integers of the form u*v*(u^2 - v^2) where 2uv and u^2 - v^2 are coprime or, alternatively, where u, v are coprime and one of them is even.
%F a(n) = 6*A258151(n). - _M. F. Hasler_, Jan 20 2019
%e 6 is in the sequence because it is the area of the 3-4-5 triangle.
%e a(7) = 210 corresponds to the two primitive Pythagorean triangles (21, 20, 29) and (35, 12, 37). See A024406. - _Wolfdieter Lang_, Jun 14 2015
%t nn = 22; (* nn must be even *) t = Union[Flatten[Table[If[GCD[u, v] == 1 && Mod[u, 2] + Mod[v, 2] == 1, u v (u^2 - v^2), 0], {u, nn}, {v, u - 1}]]]; Select[Rest[t], # < nn (nn^2 - 1) &] (* _T. D. Noe_, Sep 19 2013 *)
%Y Cf. A009111, A009112, A024406 (with multiplicity), A258150, A024407, A258151 (terms divided by 6).
%Y Subsequence of A073120 and A147778.
%K nonn,easy
%O 1,1
%A _David W. Wilson_
%E Additional comments _James R. Buddenhagen_, Aug 10 2008 and from _Max Alekseyev_, Nov 12 2008
%E Edited by _N. J. A. Sloane_, Nov 20 2008 at the suggestion of _R. J. Mathar_
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