A290451 -id:A290451 - cp's OEIS frontend

A098030 Areas of integer-sided triangles whose area equals their perimeter.

24, 30, 36, 42, 60

Offset: 1

Lekraj Beedassy, Sep 10 2004

There are no further terms. Note that without the condition "integer-sided" there are other solutions, such as (9/2, 20, 41/2) which has perimeter and area 45. - David Wasserman, Jan 03 2008

			The areas or perimeters 24, 30, 36, 42, 60 pertain respectively to triangles with sides (6, 8, 10), (5, 12, 13), (9, 10, 17), (7, 15, 20), (6, 25, 29).

S. Ainley, Mathematical Puzzles, Problem J8 p. 113, G. Bell & Sons Ltd, London (1977).

James Grime and Brady Haran, Superhero Triangles, Numberphile video (2020)

A row of the triangle in A290451.

m0 = 10 (* = initial max side *); okQ[{x_, y_, z_}] := x <= y <= z && (-x + y + z) (x + y - z) (x - y + z) (x + y + z) == 16 (x + y + z)^2; Clear[f];
f[m_] := f[m] = Select[Tuples[Range[m], 3], okQ]; f[m = m0]; f[m = 2 m]; While[f[m] != f[m/2], m = 2 m]; sides = f[m]; Total /@ sides // Sort (* Jean-François Alcover, Jul 21 2017 *)

A289218 Areas of integer-sided triangles whose area equals twice their perimeter.

Original entry on oeis.org

84, 96, 108, 120, 132, 144, 156, 168, 180, 240, 264, 300, 324, 396, 420, 684, 1224

Offset: 1

Zhining Yang, Jun 28 2017

There are no further terms.

One term, 168, corresponds to exactly two different triangles, namely [14, 30, 40] and [10, 35, 39], both with perimeter 84. The remaining terms correspond to unique triangles. - Jeppe Stig Nielsen, Mar 04 2020

			The areas 84,96,108,120,132, ... pertain respectively to triangles with sides (13,14,15), (12,16,20), (15,15,24), (10,24,26), (11,25,30), ..., equal twice their perimeter 42,48,54,60,66,...

Cf. A098030, A007237.
2nd row of the irregular triangle in A290451.
Cf. A332922.

f[a_, b_, c_] := Block[{P = Total[{a, b, c}]/2}, Sqrt[P (P - a) (P - b) (P - c)]]; Sort@ Map[f @@ # &, Select[Union@ Map[Sort, Tuples[Range@ 200, {3}]], f @@ # == 4 Total@ # &] ] (* Michael De Vlieger, Jul 03 2017 *)

Duplicate term 168 (previous a(9)) removed by Jeppe Stig Nielsen, Mar 04 2020

A289219 Areas of integer-sided triangles whose area equals 3 times their perimeter.

Original entry on oeis.org

192, 204, 210, 216, 240, 252, 264, 270, 324, 330, 336, 378, 384, 408, 420, 456, 462, 480, 504, 522, 540, 546, 624, 690, 714, 780, 792, 840, 876, 966, 990, 1176, 1248, 1320, 1380, 1806, 2394, 2460, 3120, 4446, 8436

Offset: 1

Zhining Yang, Jun 28 2017

There are no further terms.
For a(3)=210, there are 2 solutions (20,21,29),(17,25,28);
For a(11)=336, there are 2 solutions (14,48,50),(24,35,53);
For a(16)=456, a(22)=546, there are 2 solutions respectively too.

			The areas 192,204,210,216,240, ... pertain respectively to triangles with sides (20,20,24), (17,25,26), (20,21,29), (18,24,30), (16,30,34), ..., equal 3 times their perimeter 64,68,70,72,80, ...

Cf. A098030, A289218.

A row of the triangle in A290451.

f[a_, b_, c_] := Block[{P = Total[{a, b, c}]/2}, Sqrt[P (P - a) (P - b) (P - c)]]; Sort@ Map[f @@ # &, Select[Union@ Map[Sort, Tuples[Range@ 150, {3}]], f @@ # == 3 Total@# &] ] (* Michael De Vlieger, Jul 03 2017 *)

A289220 Areas of integer-sided triangles whose area equals 4 times their perimeter.

Original entry on oeis.org

336, 360, 384, 432, 456, 480, 528, 576, 624, 672, 720, 840, 960, 1056, 1176, 1200, 1224, 1296, 1584, 1680, 1944, 2064, 2088, 2184, 2328, 2520, 2736, 2856, 3240, 3696, 4440, 4488, 4896, 5256, 6600, 7728, 9240, 9360, 9384, 17688, 34320

Offset: 1

Zhining Yang, Jun 28 2017

There are no further terms.
For a(10)=672, there are 2 solutions: (28,60,80), (20,70,78).
For a(12)=840, there are 3 solutions: (35,73,102), (25,84,101), (21,89,100).

			The areas 336,360,384,432,456, ... pertain respectively to triangles with sides (26,28,30), (25,29,36), (24,32,40), (30,30,48), (25,38,51), ..., equal 4 times their perimeter 84,90,96,108,114,...

Cf. A098030, A289218, A289219.

A row of the triangle in A290451.

f[a_, b_, c_] := Block[{P = Total[{a, b, c}]/2}, Sqrt[P (P - a) (P - b) (P - c)]]; Sort@ Map[f @@ # &, Select[Union@ Map[Sort, Tuples[Range@ 200, {3}]], f @@ # == 4 Total@ # &] ] (* Michael De Vlieger, Jul 03 2017 *)

A289155 Smallest area of triangle with integer sides and area = n times perimeter.

Original entry on oeis.org

24, 84, 192, 336, 540, 756, 1134, 1344, 1710, 2100, 2640, 3000, 4056, 4116, 4680, 5376, 6936, 6804, 8664, 8400, 9240, 10164, 12696, 12000, 13500, 14196, 15390, 16296, 20184, 18720, 23064, 21504, 23232, 24276, 26040, 27000, 32856, 30324

Offset: 1

Zhining Yang, Jun 26 2017

nonn

			For n = 4, a(4)=336 means for the smallest triangle (a,b,c) = (26,28,30), the area is 336, which is 4 times the perimeter 84.

Ray Chandler, Table of n, a(n) for n = 1..5000 (first 300 terms from Zhining Yang)

Cf. A007237, A120572, A188158, A228383.

a(n) is the leading entry in row n of the triangle in A290451.

for(k=1, 50, n=0;A=10^9; d=4*k^2; e=3*d; for(b=1, sqrt(e), for (c=2*k, e/b, if(b*c>d&&c>=b, f = (b + c)*d / (b * c - d); if(f>=c, a=floor(f); if(a==f, n++; s=2*(a+b+c)*k;if(s

a(n) = A120572(2n). - Ray Chandler, Jul 27 2017

A289221 Areas of integer-sided triangles whose area equals 5 times their perimeter.

Original entry on oeis.org

540, 600, 630, 660, 750, 810, 840, 900, 930, 1050, 1080, 1320, 1380, 1500, 1560, 1590, 1740, 2040, 2070, 2280, 2310, 2520, 2580, 2970, 3150, 3240, 3720, 4020, 4350, 4530, 4620, 5460, 6270, 6300, 7260, 7560, 7800, 7980, 11730, 12210, 14040, 18870, 22260, 27030, 27300, 52530, 103020

Offset: 1

Zhining Yang, Jun 28 2017

There are no further terms.

			The areas 540,600,630,660,750, ... pertain respectively to triangles with sides (30,39,39), (30,40,50), (28,45,53), (26,51,55), (25,60,65)...., equal 5 times their perimeter 108,120,126,132,150,...

Cf. A098030, A289218, A289219, A289220.

A row of the triangle in A290451.

A289253 Areas of integer-sided triangles whose area equals 7 times their perimeter.

Original entry on oeis.org

1134, 1176, 1344, 1386, 1470, 1596, 1680, 1764, 1848, 1890, 2016, 2058, 2184, 2310, 2394, 2520, 2604, 2856, 2940, 3024, 3360, 3696, 3780, 3864, 4032, 4242, 4368, 4536, 4830, 5292, 5544, 5712, 6006, 6090, 6216, 6258, 6510, 6636, 6720

Offset: 1

Zhining Yang, Jun 29 2017

			The areas 1134,1176,1344,1386,1470, ... pertain respectively to triangles with sides (39,60,63), (42,56,70), (40,68,84), (36,77,85), (35,84,91)...., equal 7 times their perimeter 162,168,192,198,210,...

Zhining Yang, Table of n, a(n) for n = 1..77

Cf. A098030, A289218, A289219, A289220, A289221.
A row of the triangle in A290451.
Cf. A332927 (listing distinct triangles with identical areas separately).

A332879 Areas of integer-sided triangles whose area equals 6 times their perimeter.

Original entry on oeis.org

756, 768, 780, 816, 840, 864, 924, 960, 972, 984, 1008, 1020, 1056, 1080, 1092, 1116, 1140, 1188, 1260, 1296, 1320, 1344, 1380, 1404, 1500, 1512, 1536, 1620, 1632, 1680, 1716, 1740, 1824, 1836, 1848, 1920, 1980, 2016, 2088, 2160, 2184, 2244, 2376, 2436, 2460

Offset: 1

Hugo Pfoertner, Mar 02 2020

Hugo Pfoertner, Table of n, a(n) for n = 1..127, providing the full sequence.

6th row in A290451.
Cf. A098030, A289155, A289218, A289219, A289220, A289221, A289253.
Cf. A332926 (listing distinct triangles with identical areas separately).

A332689 Number of distinct areas of integer-sided triangles whose area equals n times their perimeter.

Original entry on oeis.org

5, 17, 41, 41, 47, 127, 77, 81, 171, 132, 99, 283, 94, 205, 349, 158, 115, 457, 122, 296, 530, 267, 134, 546, 219, 260, 428, 471, 130, 953, 144, 264, 613, 332, 557, 1031, 139, 346, 614, 600, 162, 1381, 169, 562, 1132, 348, 186, 1000, 363, 593, 688, 571, 164, 1123

Offset: 1

Jeppe Stig Nielsen, Feb 19 2020

nonn

Gives the row lengths of the irregular array A290451.

			For n = 2, there are 18 different (noncongruent) Heronian triangles whose area equals twice their perimeter, so A007237(2) = 18. However, two of those 18 triangles share the area 168. So there are only 17 distinct areas. Therefore, a(2) = 17.

James Grime and Brady Haran, Superhero Triangles, Numberphile video (2020).

Cf. A007237, A290451.

a[k_] := Block[{v={},r,s,t}, Do[ If[r <= s && 4 k^2 < r s <= 12 k^2 && IntegerQ[t = 4 k^2 (r + s)/(r s - 4 k^2)] && t >= s, AppendTo[v, r + s + t]], {r, Floor[2 Sqrt[3] k]}, {s, Floor[4 k^2/r], Ceiling[12 k^2/r]}]; Length@ Union@ v]; Array[a, 20] (* Giovanni Resta, Mar 04 2020 *)

from math import sqrt
def A332689(n):
    L = []; k = 4*n*n
    for x in range(1, int(2*sqrt(3)*n) + 1):
        for y in range(max(int(k/x) + 1, x), int((k + 2*n*sqrt(k + x*x))/x) + 1):
            if k*(x+y)%(x*y-k) == 0:
                s = x + y + k*(x+y)//(x*y-k)
                if s not in L: L.append(s)
    return len(L)  # Ya-Ping Lu, Dec 28 2023

a(8)-a(54) from Giovanni Resta, Mar 04 2020

A348143 Areas of integer-sided cyclic quadrilaterals whose area equals their perimeter.

Original entry on oeis.org

16, 18, 20, 30

Offset: 1

Frank M Jackson, Oct 02 2021

There are no further terms. Note that without the condition "integer-sided" there are other solutions, such as (1, 17/2, 17/2, 16) which has perimeter and area 34.

			The areas or perimeters 16, 18, 20, 30 pertain respectively to cyclic quadrilaterals with sides (4, 4, 4, 4), (3, 3, 6, 6), (2, 5, 5, 8), (5, 5, 6, 14).

Ralph H. Buchholz and James A. MacDougall, Cyclic Polygons with Rational Sides and Area, 2001. JNT 128 (2008) 17-48

Cf. A098030, A290451. First four terms of A161874.

lst={}; Do[s=(a+b+c+d)/2; If[s>a, (K=Sqrt[(s-a)(s-b)(s-c)(s-d)]; If[IntegerQ[K]&&K==2s, AppendTo[lst, Sort@{a,b,c,d}]])], {a, 1, 15}, {b, 1, a}, {c, 1, b}, {d, 1, c}]; lst

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A098030 Areas of integer-sided triangles whose area equals their perimeter.

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A289218 Areas of integer-sided triangles whose area equals twice their perimeter.

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A289219 Areas of integer-sided triangles whose area equals 3 times their perimeter.

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A289220 Areas of integer-sided triangles whose area equals 4 times their perimeter.

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A289155 Smallest area of triangle with integer sides and area = n times perimeter.

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A289221 Areas of integer-sided triangles whose area equals 5 times their perimeter.

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A289253 Areas of integer-sided triangles whose area equals 7 times their perimeter.

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A332879 Areas of integer-sided triangles whose area equals 6 times their perimeter.

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A332689 Number of distinct areas of integer-sided triangles whose area equals n times their perimeter.

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