A229159 -id:A229159 - cp's OEIS frontend

A229746 Integer areas of integer-sided triangles where two sides are of prime length.

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

Offset: 1

Michel Lagneau, Sep 28 2013

nonn

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  *
....................

			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.

Eric W. Weisstein, Heron's Formula

Cf. A188158, A229159.

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 *)

Extended by T. D. Noe, Sep 30 2013

Missing term 2970 from Giovanni Resta, Mar 08 2017

A227112 Numbers n such that there exist two primes p and q where the area A of the triangle of sides (n, p, q) is an integer.

Original entry on oeis.org

4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 26, 30, 38, 40, 42, 50, 56, 60, 68, 70, 78, 80, 90, 96, 100, 102, 104, 110, 120, 130, 144, 148, 150, 156, 160, 170, 174, 180, 182, 198, 210, 224, 234, 240, 286, 290, 300, 312, 350, 360, 370, 390, 400, 440, 510, 520, 548

Offset: 1

Michel Lagneau, Oct 02 2013

nonn

n is an even composite number because the perimeter of the triangle (n, p, q) is always even. The corresponding areas are {6, 12, 12, 60, 30, 120, 360, 66, ...}
The area is given by Heron's formula A = sqrt(s(s-n)(s-p)(s-q)) where the semiperimeter s = (n + p + q)/2.
The following table gives the first values (n, A, p, q).
+----+-----+----+----+
|  n |  A  |  p |  q |
+----+-----+----+----+
|  4 |   6 |  3 |  5 |
|  6 |  12 |  5 |  5 |
|  8 |  12 |  5 |  5 |
| 10 |  60 | 13 | 13 |
| 12 |  30 |  5 | 13 |
| 16 | 120 | 17 | 17 |
| 18 | 360 | 41 | 41 |
| 20 |  66 | 11 | 13 |
...

			12 is in the sequence because the triangle (12, 5, 13) => semiperimeter s = (12+5+13)/2 = 15, and A = sqrt(15*(15-12)*(15-5)*(15-13))= 30, with 5 and 13 prime numbers.

Cf. A229159, A229746.

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 = 300; 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, e]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* program from T. D. Noe adapted for this sequence - see A229746 *)

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A229746 Integer areas of integer-sided triangles where two sides are of prime length.

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