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 A226453 Integer areas of integer-sided triangles where at least one side is of prime length. 3
 6, 12, 24, 30, 36, 42, 60, 66, 72, 84, 90, 114, 120, 126, 132, 156, 180, 204, 210, 216, 234, 240, 252, 264, 270, 288, 300, 306, 330, 336, 360, 390, 396, 420, 456, 462, 504, 510, 522, 528, 546, 570, 624, 630, 660, 684, 690, 714, 720, 756, 780, 798, 840, 864 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Subset of A188158. 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. There exist triangles whose two distinct integer sides are a prime number, for example: a(n) = 6 with sides (3,4,5); a(n) = 30 with sides (5,12,13); a(n) = 66 with sides (11,13,20); a(n) = 72 with sides (5,29,30); a(n) = 114 with sides (19,20,37). The following table gives the first values (A, a, b, c): ********************** *  A * a  * b  *  c  * ********************** *  6 *  3 *  4 *  5  * * 12 *  5 *  5 *  6  * * 12 *  5 *  5 *  8  * * 24 *  4 * 13 * 15  * * 30 *  5 * 12 * 13  * * 36 *  3 * 25 * 26  * * 36 *  9 * 10 * 17  * * 42 *  7 * 15 * 20  * * 60 *  6 * 25 * 29  * * 60 *  8 * 15 * 17  * * 60 * 10 * 13 * 13  * * 60 * 13 * 13 * 24  * * 66 * 11 * 13 * 20  * * 72 *  5 * 29 * 30  * ...................... LINKS Giovanni Resta, Table of n, a(n) for n = 1..10000 EXAMPLE 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. MATHEMATICA nn=1000; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0

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Last modified May 7 19:23 EDT 2021. Contains 343652 sequences. (Running on oeis4.)