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%I #31 May 30 2023 06:53:07
%S 6,12,24,30,36,42,60,66,72,84,90,114,120,126,132,156,180,204,210,216,
%T 234,240,252,264,270,288,300,306,330,336,360,390,396,420,456,462,504,
%U 510,522,528,546,570,624,630,660,684,690,714,720,756,780,798,840,864
%N Integer areas of integer-sided triangles where at least one side is of prime length.
%C Subset of A188158.
%C The area of the triangles (a,b,c) are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
%C There exist triangles where two distinct integer sides are a prime number, for example:
%C a(n) = 6 with sides (3,4,5);
%C a(n) = 30 with sides (5,12,13);
%C a(n) = 66 with sides (11,13,20);
%C a(n) = 72 with sides (5,29,30);
%C a(n) = 114 with sides (19,20,37).
%C The following table gives the first values (A, a, b, c):
%C **********************
%C * A * a * b * c *
%C **********************
%C * 6 * 3 * 4 * 5 *
%C * 12 * 5 * 5 * 6 *
%C * 12 * 5 * 5 * 8 *
%C * 24 * 4 * 13 * 15 *
%C * 30 * 5 * 12 * 13 *
%C * 36 * 3 * 25 * 26 *
%C * 36 * 9 * 10 * 17 *
%C * 42 * 7 * 15 * 20 *
%C * 60 * 6 * 25 * 29 *
%C * 60 * 8 * 15 * 17 *
%C * 60 * 10 * 13 * 13 *
%C * 60 * 13 * 13 * 24 *
%C * 66 * 11 * 13 * 20 *
%C * 72 * 5 * 29 * 30 *
%C ......................
%H Giovanni Resta, <a href="/A226453/b226453.txt">Table of n, a(n) for n = 1..10000</a>
%e 114 is in the sequence because the triangle (19, 20, 37) => semiperimeter s = (19+20+37)/2 = 38, and A = sqrt(38*(38-19)*(38-20)*(38-37)) = 114.
%t nn=1000; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0<area2<=nn^2 && (PrimeQ[a] || PrimeQ[b] || PrimeQ[c]) && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
%Y Cf. A188158.
%K nonn
%O 1,1
%A _Michel Lagneau_, Sep 16 2013
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