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A156688 The total number of distinct Pythagorean triples with an area numerically equal to n times their perimeters.

%I A156688 #18 Jul 22 2025 06:21:16
%S A156688 2,3,6,4,6,9,6,5,10,9,6,12,6,9,18,6,6,15,6,12,18,9,6,15,10,9,14,12,6,
%T A156688 27,6,7,18,9,18,20,6,9,18,15,6,27,6,12,30,9,6,18,10,15,18,12,6,21,18,
%U A156688 15,18,9,6,36,6,9,30,8,18,27,6,12,18,27,6,25,6,9,30,12,18,27,6,18,18,9,6,36,18,9,18,15,6,45,18,12,18,9,18
%N A156688 The total number of distinct Pythagorean triples with an area numerically equal to n times their perimeters.
%C A156688 The members of this sequence are also 1/2 the number of divisors of 8n^2. The corresponding results for primitive triangles only are in A068068.
%C A156688 Also, the total number of distinct "areas with equal border", that is: Let x, y be positive integers so that the area xy equals the border around it with thickness n. As a formula it is: 2xy = (x+2n)(y+2n). To compare with the original, the areas at thickness 5 are 11x210, 12x110, 14x60, 15x50, 18x35, 20x30. - Juhani Heino, Jul 22 2012
%D A156688 Chi, Henjin and Killgrove, Raymond; Problem 1447, Crux Math 15(5), May 1989.
%D A156688 Chi, Henjin and Killgrove, Raymond; Solution to Problem 1447, Crux Math 16(7), September 1990.
%H A156688 Antti Karttunen, <a href="/A156688/b156688.txt">Table of n, a(n) for n = 1..65537</a>
%H A156688 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Right-angled Triangles and Pythagoras' Theorem</a>
%F A156688 a(n) = A000005(8n^2)/2 = A078644(2n).
%e A156688 There are 6 Pythagorean triples whose area is 5 times their perimeters - (21,220,221), (22,120,122), (24,70,74), (25,60,65),(28,45,53) and (30,40,50) - hence a(5)=6.
%t A156688 1/2 DivisorSigma[0,8#^2] &/@Range[75]
%o A156688 (PARI) A156688(n) = (numdiv(8*n*n)/2); \\ _Antti Karttunen_, Sep 27 2018
%Y A156688 Cf. A000005, A068068.
%K A156688 easy,nice,nonn
%O A156688 1,1
%A A156688 _Ant King_, Feb 18 2009
%E A156688 More terms from _Antti Karttunen_, Sep 27 2018