%I #50 Jan 05 2025 19:51:35

%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="https://web.archive.org/web/2024*/https://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 *)

%o (PARI) select( {is_A024365(n)=my(N=1+#n=divisors(2*n)); for(i=1, N\2, gcd(n[i], n[N-i])==1 && issquare(n[i]^2+n[N-i]^2) && return(n[i]))}, [1..10^4]) \\ is_A024365 returns the smaller leg if n is a term, else 0. - _M. F. Hasler_, Jun 06 2024

%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_