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A229746 Integer areas of integer-sided triangles where two sides are of prime length. 3
6, 12, 30, 60, 66, 72, 114, 120, 180, 210, 240, 330, 336, 360, 396, 420, 456, 660, 756, 780, 840, 900, 984, 1116, 1200, 1248, 1260, 1290, 1320, 1584, 1590, 1680, 1710, 1716, 1770, 1800, 1980, 2100, 2310, 2400, 2460, 2496, 2520, 2604, 2640, 2940, 2970, 3060, 3120 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Subset of A188158. The length of the third side is an even composite number because the perimeter is always even.

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.

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  *

*  30 *  5 * 12 * 13  *

*  60 * 10 * 13 * 13  *

*  66 * 11 * 13 * 20  *

*  72 *  5 * 29 * 30  *

* 114 * 19 * 20 * 37  *

* 120 * 16 * 17 * 17  *

* 120 * 17 * 17 * 30  *

* 180 * 13 * 30 * 37  *

....................

LINKS

Table of n, a(n) for n=1..49.

Eric W. Weisstein, Heron's Formula

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, with 19 and 37 prime numbers.

MATHEMATICA

area[a_, b_, c_] := Module[{s = (a + b + c)/2, a2}, a2 = s (s - a) (s - b) (s - c); If[a2 < 0, 0, Sqrt[a2]]]; goodQ[a_, b_, c_] := Module[{ar = area[a, b, c]}, ar > 0 && IntegerQ[ar]]; nn = 80; t = {}; ps = Prime[Range[2, nn]]; mx = 3*ps[[-1]]; Do[If[p <= q && goodQ[p, q, e], aa = area[p, q, e]; If[aa <= mx, AppendTo[t, aa]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* T. D. Noe, Oct 01 2013 *)

CROSSREFS

Cf. A188158, A229159.

Sequence in context: A125056 A011987 A036690 * A256579 A322374 A014131

Adjacent sequences:  A229743 A229744 A229745 * A229747 A229748 A229749

KEYWORD

nonn

AUTHOR

Michel Lagneau, Sep 28 2013

EXTENSIONS

Extended by T. D. Noe, Sep 30 2013

Missing term 2970 from Giovanni Resta, Mar 08 2017

STATUS

approved

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Last modified April 13 11:24 EDT 2021. Contains 342936 sequences. (Running on oeis4.)