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.

Previous Showing 11-20 of 23 results. Next

A334665 Perimeters of Heronian triangles with mutually distinct side lengths.

Original entry on oeis.org

12, 24, 30, 32, 36, 40, 42, 44, 48, 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

Wesley Ivan Hurt, May 07 2020

Keywords

Examples

			a(1) = 12; there is one Heronian triangle with perimeter 12 such that all side lengths are mutually distinct, [3,4,5].
a(4) = 32; there is one Heronian triangle with perimeter 32 such that all side lengths are mutually distinct, [4,13,15].

Links

Crossrefs

Cf. A051516, A051518.

A331366 Number of Heronian triangles with mutually distinct side lengths and perimeter A334665(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 1, 2, 3, 1, 1, 3, 8, 1, 4, 4, 3, 1, 2, 10, 1, 5, 1, 5, 3, 6, 1, 9, 2, 5, 6, 6, 2, 3, 5, 6, 8, 1, 10, 1, 1, 1, 6, 8, 3, 2, 2, 12, 8, 4, 6, 5, 4, 11, 14, 5, 1, 11, 7, 3, 1, 6, 1, 2, 14, 11, 1, 9, 14, 11, 1, 4, 11, 3, 1, 11, 4, 2, 7, 4, 12
Offset: 1

Views

Author

Wesley Ivan Hurt, May 07 2020

Keywords

Examples

			a(1) = 1; there is one Heronian triangle with perimeter A334665(1) = 12 such that all side lengths are mutually distinct, [3,4,5].
a(4) = 1; there is one Heronian triangle with perimeter A334665(4) = 32 such that all side lengths are mutually distinct, [4,13,15].

Links

Crossrefs

Cf. A051516, A051518, A334665.

A334936 Perimeters of Heronian triangles with even side lengths.

Original entry on oeis.org

24, 32, 36, 48, 60, 64, 72, 80, 84, 88, 96, 100, 108, 112, 120, 128, 132, 136, 140, 144, 152, 156, 160, 168, 176, 180, 192, 196, 200, 208, 216, 220, 224, 228, 240, 252, 256, 260, 264, 272, 280, 288, 300, 304, 308, 312, 320, 324, 328, 336, 340, 344, 348, 352, 360, 364
Offset: 1

Views

Author

Wesley Ivan Hurt, May 16 2020

Keywords

Examples

			a(1) = 24; there is one Heronian triangle with even side lengths whose perimeter is 24, [6,8,10].
a(2) = 32; there is one Heronian triangle with even side lengths whose perimeter is 32, [10,10,12].

Links

Crossrefs

Cf. A051518.

A305703 Positive even integers that are not the perimeter of any Heronian triangle.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 20, 22, 26, 28, 34, 38, 46, 52, 58, 62, 74, 82, 86, 92, 94, 102, 106, 116, 118, 122, 124, 134, 138, 142, 146, 148, 158, 166, 178, 184, 188, 194, 202, 206, 212, 214, 218, 226, 244, 246, 254, 262, 274, 278, 282, 284, 298, 302, 314, 316
Offset: 1

Views

Author

Peter Kagey, Jun 08 2018

Keywords

Comments

Equivalently, this sequence is all positive even numbers that are not in A051518.
Contains all even semiprimes (A100484).

Links

Crossrefs

Cf. A051518, A100484.

A305704 Numbers that are the semiperimeter of some triangle with integer sides and area.

Original entry on oeis.org

6, 8, 9, 12, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 70, 72, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 95, 96, 98, 99, 100, 102, 104, 105
Offset: 1

Views

Author

Peter Kagey, Jun 08 2018

Keywords

Links

Crossrefs

Cf. A051518.

Programs

  • Mathematica
    hp[p_] := Catch@ Block[{c, q=p/2}, Do[c = p-a-b; If[IntegerQ@ Sqrt[q (q-a) (q-b) (q-c)], Throw@ True], {a, p/2-1}, {b, p/2-a+1, p/2-1}]; False]; Select[Range@ 105, hp[2 #] &] (* Giovanni Resta, Jun 14 2018 *)

Formula

a(n) = A051518(n)/2.

A334983 Perimeters of Heronian triangles where the lengths of the smallest and largest sides are coprime.

Original entry on oeis.org

12, 16, 18, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 90, 96, 98, 100, 104, 108, 110, 112, 114, 120, 126, 128, 130, 132, 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

Wesley Ivan Hurt, May 18 2020

Keywords

Comments

This sequence includes the perimeters of all primitive Heronian triangles (A096468). First differs from A096468 at a(38) = 140.

Examples

			a(1) = 12; there is one Heronian triangle with perimeter 12, which is [3,4,5] and the lengths of the smallest and largest sides are coprime (GCD(3,5) = 1).
a(5) = 32; there is one Heronian triangle with perimeter 32, [4,13,15] and the lengths of the smallest and middle sides are coprime (GCD(4,13) = 1).

Links

Crossrefs

Cf. A051518, A096468, A334984, A334989.

A334984 Perimeters of Heronian triangles where the lengths of the middle and largest sides are coprime.

Original entry on oeis.org

12, 16, 18, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 68, 70, 72, 76, 78, 80, 84, 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, 176, 180, 182, 186, 190, 192, 196, 198, 200, 204, 208
Offset: 1

Views

Author

Wesley Ivan Hurt, May 18 2020

Keywords

Comments

This sequence includes the perimeters of all primitive Heronian triangles (A096468). First differs from A096468 at a(16) = 68.

Examples

			a(1) = 12; there is one Heronian triangle with perimeter 12, which is [3,4,5] and the lengths of the middle and largest sides are coprime (GCD(4,5) = 1).
a(5) = 32; there is one Heronian triangle with perimeter 32, [4,13,15] and the lengths of the smallest and middle sides are coprime (GCD(4,13) = 1).

Links

Crossrefs

Cf. A051518, A096468, A334983, A334989.

A334989 Perimeters of Heronian triangles where the lengths of the smallest and middle sides are coprime.

Original entry on oeis.org

12, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 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

Wesley Ivan Hurt, May 18 2020

Keywords

Comments

This sequence includes the perimeters of all primitive Heronian triangles (A096468). First differs from A096468 at a(2) = 30.

Examples

			a(1) = 12; there is one Heronian triangle with perimeter 12, which is [3,4,5] and the lengths of the smallest and middle sides are coprime (GCD(3,4) = 1).
a(3) = 32; there is one Heronian triangle with perimeter 32, [4,13,15] and the lengths of the smallest and middle sides are coprime (GCD(4,13) = 1).

Links

Crossrefs

Cf. A051518, A096468, A334983, A334984.

A305705 Positive integers which are not the semiperimeter of any triangle with integer sides and area.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 11, 13, 14, 17, 19, 23, 26, 29, 31, 37, 41, 43, 46, 47, 51, 53, 58, 59, 61, 62, 67, 69, 71, 73, 74, 79, 83, 89, 92, 94, 97, 101, 103, 106, 107, 109, 113, 122, 123, 127, 131, 137, 139, 141, 142, 149, 151, 157, 158, 163, 167, 173, 179, 181
Offset: 1

Views

Author

Peter Kagey, Jun 08 2018

Keywords

Links

Crossrefs

Complement of A305704.
Cf. A051518, A305703.

Formula

a(n) = A305703(n)/2.

A334589 Perimeters of nonprimitive Heronian triangles.

Original entry on oeis.org

24, 32, 36, 48, 54, 60, 64, 72, 80, 84, 88, 90, 96, 100, 108, 112, 120, 126, 128, 132, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240, 250, 252, 256, 260, 264, 270, 272, 276, 280, 288, 294, 300, 304, 306
Offset: 1

Views

Author

Wesley Ivan Hurt, May 06 2020

Keywords

Examples

			a(1) = 24; there is one nonprimitive Heronian triangle with a perimeter of 24, which is [6,8,10].
a(2) = 32; there is one nonprimitive Heronian triangle with a perimeter of 32, which is [10,10,12].

Links

Crossrefs

Cf. A051518, A096468, A334590.
Previous Showing 11-20 of 23 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.