cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A334934 Largest possible odd side length of a primitive Heronian triangle with perimeter A096468(n).

Original entry on oeis.org

5, 5, 5, 13, 15, 17, 17, 15, 13, 21, 17, 25, 25, 29, 29, 25, 25, 29, 35, 37, 37, 39, 41, 41, 37, 45, 41, 51, 53, 33, 53, 51, 39, 53, 61, 41, 65, 65, 65, 65, 73, 73, 75, 75, 75, 73, 73, 77, 75, 61, 85, 87, 89, 89, 87, 87, 95, 97, 97, 97, 101, 101, 101, 97, 109, 87, 111, 113
Offset: 1

Views

Author

Wesley Ivan Hurt, May 16 2020

Keywords

Examples

			a(1) = 5; there is one primitive Heronian triangle with perimeter A096468(1) = 12, which is [3,4,5] and the largest odd side lengths is 5.
a(6) = 17; there are two primitive Heronian triangles with perimeter A096468(6) = 36, [9,10,17] and [10,13,13]. The largest odd side length of the two triangles is 17.

Links

Crossrefs

Cf. A096468, A334928.

A331209 Largest possible area of a primitive Heronian triangle with perimeter A096468(n).

Original entry on oeis.org

6, 12, 12, 30, 24, 60, 60, 84, 66, 84, 120, 126, 84, 60, 168, 132, 204, 210, 84, 114, 156, 180, 252, 360, 264, 420, 420, 468, 306, 330, 504, 456, 420, 630, 720, 780, 660, 408, 630, 660, 1020, 456, 924, 1170, 840, 1260, 984, 924, 1020, 1290, 522, 1320, 420, 1092, 1116, 1710
Offset: 1

Views

Author

Wesley Ivan Hurt, May 03 2020

Keywords

Examples

			a(1) = 6; there is one primitive Heronian triangle with perimeter A096468(1) = 12, which is [3,4,5] and its area is 6.
a(6) = 60; there are two primitive Heronian triangles with perimeter A096468(6) = 36, [9,10,17] and [10,13,13] with areas 36 and 60. The largest of these is 60.

Links

Crossrefs

Cf. A096468.

A334970 Sum of the values of the "shortest + middle - longest" side lengths of all primitive Heronian triangles with perimeter A096468(n).

Original entry on oeis.org

2, 4, 2, 4, 2, 12, 6, 14, 4, 6, 18, 18, 6, 2, 22, 6, 16, 26, 2, 2, 4, 8, 38, 36, 8, 88, 46, 26, 20, 6, 44, 12, 8, 20, 72, 30, 64, 6, 28, 44, 66, 6, 50, 62, 24, 160, 18, 14, 20, 36, 4, 72, 2, 34, 28, 70, 142, 164, 48, 100, 10, 36, 82, 94, 86, 48, 124, 122, 46, 24, 108, 18, 34
Offset: 1

Views

Author

Wesley Ivan Hurt, May 17 2020

Keywords

Comments

a(n) is the excess length on the shortest and/or middle sides of all the primitive Heronian triangles with perimeter A096468(n) to avoid becoming degenerate triangles whose longest side is equal to the longest side of each primitive Heronian triangle.

Examples

			a(1) = 2; there is one primitive Heronian triangle with perimeter A096468(1) = 12, which is [3,4,5] and 3 + 4 - 5 = 2.
a(6) = 12; there are two primitive Heronian triangles with perimeter A096468(6) = 36, [9,10,17] and [10,13,13]. Then (9 + 10 - 17) + (10 + 13 - 13) = 2 + 10 = 12.

Links

Crossrefs

Cf. A096468.

A051518 Perimeters of Heronian triangles.

Original entry on oeis.org

12, 16, 18, 24, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 96, 98, 100, 104, 108, 110, 112, 114, 120, 126, 128, 130, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174, 176, 180, 182, 186, 190, 192, 196, 198, 200, 204
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

A triangle with integer sides and area is also called a Heronian triangle, and every such triangle can be positioned in the plane so that its three vertices have integer coordinates. - Peter Kagey, Jan 24 2018

Links

Crossrefs

Cf. A096468, A298079.
Positions of nonzero values in A051516.

Extensions

Name changed by Wesley Ivan Hurt, May 16 2020

A070138 Number of integer triangles with an integer area having relatively prime sides a, b and c such that a + b + c = n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 3
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

From Peter Kagey, Jan 30 2018: (Start)
a(k) > 0 if and only if k is in A096468.
Records appear at indices 12, 36, 54, 84, 98, 162, 242, 338, 484, 578, ....
a(2k - 1) = 0 for all integers k > 0.
(End)

Links

Crossrefs

Cf. A051493, A051516, A070088, A070109, A070143, A096468, A239246.

Extensions

Corrected by T. D. Noe, Jun 17 2004

A330912 Sum of the smallest side lengths of all Heronian triangles with perimeter A051518(n).

Original entry on oeis.org

3, 5, 5, 6, 5, 14, 38, 8, 20, 11, 37, 29, 43, 7, 31, 64, 11, 17, 37, 84, 19, 15, 70, 130, 22, 87, 101, 133, 122, 38, 241, 25, 149, 25, 111, 123, 225, 39, 220, 54, 120, 327, 254, 57, 103, 162, 227, 371, 41, 321, 34, 43, 29, 278, 373, 76, 70, 95, 577, 567, 157, 476, 221
Offset: 1

Views

Author

Wesley Ivan Hurt, May 02 2020

Keywords

Examples

			a(1) = 3; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its smallest side length is 3.
a(6) = 14; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum is 4 + 10 = 14.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330915, A330916.

Formula

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)))) * k, where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A330915 Sum of the "middle" side lengths (b such that a <= b <= c) of all Heronian triangles with perimeter A051518(n).

Original entry on oeis.org

4, 5, 5, 8, 12, 23, 45, 15, 29, 13, 48, 30, 77, 24, 69, 117, 25, 25, 46, 119, 20, 26, 110, 246, 26, 167, 172, 205, 169, 79, 468, 33, 229, 38, 222, 167, 429, 41, 429, 101, 270, 560, 416, 100, 153, 276, 390, 717, 50, 615, 61, 61, 60, 404, 634, 214, 130, 130, 1033, 975, 382
Offset: 1

Views

Author

Wesley Ivan Hurt, May 02 2020

Keywords

Examples

			a(1) = 4; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its "middle" side length is 4.
a(6) = 23; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum is 13 + 10 = 23.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330916.

Formula

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)))) * i, where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A330916 Sum of the largest side lengths of all Heronian triangles with perimeter A051518(n).

Original entry on oeis.org

5, 6, 8, 10, 13, 27, 61, 17, 35, 20, 59, 41, 96, 25, 80, 139, 30, 26, 57, 157, 37, 37, 140, 296, 40, 196, 207, 250, 209, 91, 587, 52, 294, 51, 267, 214, 498, 50, 539, 117, 310, 697, 530, 147, 206, 342, 503, 856, 73, 744, 75, 68, 85, 550, 793, 256, 172, 155, 1270, 1202
Offset: 1

Views

Author

Wesley Ivan Hurt, May 02 2020

Keywords

Examples

			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) = 27; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum is 15 + 12 = 27.

Links

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915.

Formula

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)))) * (c(n)-i-k), where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A096467 Numbers that can be the longest side of a primitive Heronian triangle.

Original entry on oeis.org

5, 6, 8, 13, 15, 17, 20, 21, 24, 25, 26, 28, 29, 30, 35, 36, 37, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 53, 55, 56, 58, 60, 61, 63, 65, 66, 68, 69, 70, 73, 74, 75, 77, 80, 82, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 112, 113
Offset: 1

Views

Author

T. D. Noe, Jun 22 2004

Keywords

Comments

Here a primitive Heronian triangle has integer sides a,b,c with gcd(a,b,c) = 1 and integral area. Note that all primes of the form 4k+1 are in this sequence. It appears that a prime of the form 4k+3 is never the longest side of a Heronian triangle. Cheney's article contains many theorems about these triangles.

Examples

			5 is on this list because the triangle with sides 3, 4, 5 has integral area.

Links

Crossrefs

Cf. A083875 (area/6 of primitive Heronian triangles), A096468 (perimeter of primitive Heronian triangles).
Cf. A239246, A306626.

Programs

  • Mathematica
    nn=150; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s] && GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], AppendTo[lst, a]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

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

Original entry on oeis.org

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

Views

Author

Wesley Ivan Hurt, May 02 2020

Keywords

Examples

			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.

Links

Crossrefs

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

Formula

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
Previous Showing 11-20 of 35 results. Next

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