A070138 -id:A070138 - cp's OEIS frontend

A330921 Sum of the areas of all Heronian triangles with perimeter A051518(n).

6, 12, 12, 24, 30, 72, 198, 60, 126, 66, 288, 180, 360, 84, 330, 648, 132, 204, 420, 876, 114, 156, 840, 1764, 264, 1350, 1632, 2016, 1830, 624, 3816, 330, 2604, 456, 2280, 2352, 4800, 780, 4422, 1224, 2940, 7068, 5430, 912, 2310, 3744, 5520, 9144, 984, 8736, 1020

Offset: 1

Wesley Ivan Hurt, May 02 2020

nonn

			a(1) = 6; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its area is 3*4/2 = 6.
a(6) = 72; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum of their areas 24 + 48 = 72.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.

Cf. A330912, A330915, A330916.

a(n) = Sum_{k=1..floor(c(n)/3)} Sum_{i=k..floor((c(n)-k)/2)} sign(floor((i+k)/(c(n)-i-k+1))) * chi(sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k)))) * sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k))), where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A070202 Number of integer triangles with perimeter n, integer inradius and side lengths that are not relatively prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			For perimeter 24, only the triangle with a=6, b=8, c=10 has an integer inradius (2), therefore a(24)=1. The next examples are a(32)=1 with a=10, b=10, c=12 and a(36)=1 with a=9, b=12, c=15.

Eric Weisstein's World of Mathematics, Incircle.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070138, A051493, A070109, A070209.

Definition corrected by Georg Fischer, Apr 04 2024

A330917 Largest possible side length, a, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

Original entry on oeis.org

3, 5, 5, 6, 5, 10, 10, 8, 13, 11, 15, 16, 15, 7, 15, 20, 11, 17, 20, 20, 19, 15, 25, 26, 22, 25, 30, 29, 32, 25, 30, 25, 35, 25, 30, 39, 40, 39, 33, 34, 40, 45, 48, 38, 35, 51, 50, 53, 41, 52, 34, 43, 29, 55, 50, 35, 39, 57, 60, 65, 55, 64, 51, 65, 65, 60, 68, 61, 70, 65

Offset: 1

Wesley Ivan Hurt, May 03 2020

nonn

			a(1) = 3; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its shortest side is 3.
a(6) = 10; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], whose smallest side lengths are 4 and 10. The largest of these is 10.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Cf. A330923, A331199.

A330923 Largest possible side length, b, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

Original entry on oeis.org

4, 5, 5, 8, 12, 13, 13, 15, 15, 13, 17, 17, 25, 24, 25, 29, 25, 25, 25, 29, 20, 26, 30, 35, 26, 40, 39, 40, 41, 40, 51, 33, 48, 38, 50, 45, 58, 41, 60, 51, 65, 65, 61, 60, 56, 68, 65, 75, 50, 72, 61, 61, 60, 74, 80, 84, 68, 65, 87, 89, 90, 82, 87, 80, 89, 102, 100, 74

Offset: 1

Wesley Ivan Hurt, May 03 2020

nonn

			a(1) = 4; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its middle side is 4.
a(6) = 13; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], whose middle side lengths are 13 and 10. The largest of these is 13.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Cf. A330917, A331199.

A331199 Largest possible side length, c, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

Original entry on oeis.org

5, 6, 8, 10, 13, 15, 17, 17, 20, 20, 21, 24, 26, 25, 29, 30, 30, 26, 29, 35, 37, 37, 39, 41, 40, 41, 45, 48, 48, 51, 53, 52, 53, 51, 58, 60, 61, 50, 65, 65, 68, 70, 74, 74, 75, 75, 78, 80, 73, 82, 75, 68, 85, 87, 89, 89, 87, 87, 95, 97, 97, 97, 101, 102, 104, 106

Offset: 1

Wesley Ivan Hurt, May 03 2020

nonn

			a(1) = 5; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its largest side length is 5.
a(6) = 15; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], whose largest side lengths are 15 and 12. The largest of these is 15.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Cf. A330917, A330923.

A385819 Numbers k such that there are least five primitive Heron triangles having the same area and perimeter k.

Original entry on oeis.org

2842, 3542, 5642, 5750, 6314, 7238, 7546, 9790, 15470, 15778, 17710, 20026, 21658, 21970, 22610, 26962

Offset: 1

Zhining Yang, Jul 09 2025

			3542 is a term because there exists 5 primitive Heron triangles: {{421,1518,1603}, {511,1375,1656}, {583,1288,1671},{759,1096,1687}, {851,1001,1690}} with same perimeter 3542 and same area 318780.
20026 is a term because there exists 6 primitive Heron triangles: {{2108,8493,9425}, {2173,8398,9455}, {2261,8277,9488}, {2418,8075,9533}, {4123,6205,9698}, {4588,5729,9709}} with same perimeter 20026 and same area 8410920.

BBS Math Blog, Primitive Heron triangles with equal perimeter and area (in Chinese).

Cf. A051516, A070138, A188158

sol = Association[];
For[n = 2, n <= 6000, n += 2,
For[z = Ceiling[n/3], z < Floor[n/2], z++,
For[x = 1, x < Floor[n/3], x++, y = n - x - z;
   If[x + y > z > y > x && GCD[x, y, z] == 1, p = (x + y + z)/2;
    A = Sqrt[p (p - x) (p - y) (p - z)];
    If[IntegerQ[A], d = ToString@n <> "->" <> ToString@A; t = {x, y, z};
     If[KeyExistsQ[sol, d], AppendTo[sol[d], t], sol[d] = {t}]]]]]];
Select[sol, Length@# > 4 &]

A330922 Largest possible area of a Heronian triangle with perimeter A051518(n).

Original entry on oeis.org

6, 12, 12, 24, 30, 48, 60, 60, 84, 66, 108, 120, 126, 84, 150, 192, 132, 204, 210, 240, 114, 156, 300, 336, 264, 360, 432, 420, 480, 468, 540, 330, 588, 456, 600, 756, 768, 780, 726, 816, 840, 972, 1080, 456, 924, 1170, 1200, 1260, 984, 1344, 1020, 1290, 522, 1452

Offset: 1

Wesley Ivan Hurt, May 02 2020

nonn

			a(1) = 6; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its area is 3*4/2 = 6.
a(6) = 48; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], with areas 24 and 48. The largest area of the two triangles is 48.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.

Cf. A330912, A330915, A330916, A330921.

cp's OEIS Frontend

A330921 Sum of the areas of all Heronian triangles with perimeter A051518(n).

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A070202 Number of integer triangles with perimeter n, integer inradius and side lengths that are not relatively prime.

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A330917 Largest possible side length, a, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

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A330923 Largest possible side length, b, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

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A331199 Largest possible side length, c, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

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A385819 Numbers k such that there are least five primitive Heron triangles having the same area and perimeter k.

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Mathematica

A330922 Largest possible area of a Heronian triangle with perimeter A051518(n).

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