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.

Showing 1-9 of 9 results.

A070122 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an acute scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

33, 45, 53, 60, 70, 83, 90, 92, 106, 114, 119, 132, 134, 142, 148, 162, 165, 168, 175, 181, 183, 197, 200, 203, 204, 218, 224, 237, 240, 245, 247, 261, 264, 267, 268, 282, 290, 293, 296, 309, 316, 317, 319, 333, 341, 345, 348
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			70 is a term because [A070080(70), A070081(70), A070082(70)]=[5<7<8], A070084(70)=gcd(5,7,8)=1, A070085(70)=5^2+7^2-8^2=25+49-64=10>0.

Links

Crossrefs

Cf. A070096, A070118, A070110, A070112.

Programs

  • Mathematica
    m = 55 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
Position[triangles, {a_, b_, c_} /; a < b < c && GCD[a, b, c] == 1 && a^2 + b^2 - c^2 > 0] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

A070123 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an acute scalene integer triangle with prime side lengths.

Original entry on oeis.org

240, 544, 799, 911, 1262, 1568, 2621, 2681, 2856, 3369, 3648, 4246, 5194, 5541, 6576, 6626, 6725, 7441, 7503, 7565, 7902, 7944, 8882, 8956, 9332, 9452, 9472, 9888, 9988, 10421, 10498, 10502, 11075, 11079, 11622
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			240 is a term because: [A070080(240), A070081(240), A070082(240)]=[7<11<13], A070085(240)=7^2+11^2-13^2=49+121-169=1>0.

Links

Crossrefs

Cf. A070097, A070118, A070111, A070112.

Programs

  • Mathematica
    m = 500 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
Position[triangles, {a_, b_, c_} /; a < b < c && AllTrue[{a, b, c}, PrimeQ] && a^2 + b^2 - c^2 > 0] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

A070131 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

8, 13, 20, 21, 25, 29, 30, 36, 37, 41, 42, 44, 49, 56, 57, 59, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 86, 89, 96, 97, 99, 100, 101, 102, 105, 110, 111, 113, 115, 122, 123, 125, 126, 127, 128, 130, 131, 138, 141, 144, 147, 152, 153
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(20)=69: [A070080(69), A070081(69), A070082(69)]=[5<6<9], A070084(69)=gcd(5,6,9)=1, A070085(69)=5^2+6^2-9^2=25+36-81=-20<0.

Links

Crossrefs

Cf. A070104, A070110, A070127, A070112.

A070132 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle with prime side lengths.

Original entry on oeis.org

30, 101, 153, 193, 328, 334, 392, 444, 454, 519, 603, 621, 771, 777, 795, 878, 1005, 1123, 1135, 1508, 1526, 1538, 1694, 1818, 1848, 1858, 1999, 2023, 2037, 2066, 2193, 2223, 2253, 2454, 2663, 2903, 3055, 3321, 3363
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(10)=519: [A070080(519), A070081(519), A070082(519)]=[5<17<19], A070085(519)=5^2+17^2-19^2=25+289-361=-47<0.

Links

Crossrefs

Cf. A070105, A070111, A070127, A070112.

A070113 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

8, 13, 17, 20, 21, 25, 29, 30, 33, 36, 37, 41, 42, 44, 45, 49, 53, 56, 57, 59, 60, 62, 66, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 86, 89, 90, 92, 96, 97, 99, 100, 101, 102, 105, 106, 110, 111, 113, 114, 115, 119, 122, 123, 125, 126
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			36 is a term [A070080(36), A070081(36), A070082(36)]=[3<6<7], A070084(36)=gcd(3,6,7)=1.

Links

Crossrefs

Cf. A005044, A070122, A070131, A070112, A070110.

Programs

  • Mathematica
    m = 50 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
Position[triangles, {a_, b_, c_} /; a < b < c && GCD[a, b, c] == 1] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

A070114 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle with prime side lengths.

Original entry on oeis.org

30, 101, 153, 193, 240, 328, 334, 392, 444, 454, 519, 544, 603, 621, 771, 777, 795, 799, 878, 911, 1005, 1123, 1135, 1262, 1508, 1526, 1538, 1568, 1694, 1818, 1848, 1858, 1999, 2023, 2037, 2066, 2193, 2223, 2253, 2454
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(2)=101: [A070080(101), A070081(101), A070082(101)]=[5<7<11].

Links

Crossrefs

Cf. A070090, A070123, A070132, A070112, A070111.

A070130 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse scalene integer triangle.

Original entry on oeis.org

8, 13, 20, 21, 25, 29, 30, 36, 37, 41, 42, 44, 49, 50, 56, 57, 59, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 86, 87, 89, 96, 97, 99, 100, 101, 102, 105, 110, 111, 113, 115, 122, 123, 125, 126, 127, 128, 130, 131, 138, 139, 141, 143
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(20)=67: [A070080(67), A070081(67), A070082(67)]=[4<7<9], A070085(67)=4^2+7^2-9^2=16+49-81=-16<0.

Links

Crossrefs

Cf. A024156, A070132, A070131, A070112, A070130.

A070121 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute scalene integer triangle.

Original entry on oeis.org

33, 45, 53, 60, 70, 83, 90, 92, 106, 114, 119, 132, 134, 142, 148, 162, 165, 168, 175, 181, 183, 197, 200, 203, 204, 218, 221, 224, 237, 240, 245, 247, 261, 264, 267, 268, 282, 290, 293, 296, 309, 312, 316, 317, 319, 333, 341
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(4)=60: [A070080(60), A070081(60), A070082(60)]=[4<7<8], A070085(60)=4^2+7^2-8^2=16+49-64=1>0.

Links

Crossrefs

Cf. A024154, A070122, A070123, A070112, A070118.

A307894 Hypotenuses of primitive Pythagorean triangles with prime length, having the property that the sum and absolute difference of the shorter legs are both prime numbers.

Original entry on oeis.org

13, 17, 37, 53, 73, 97, 109, 113, 137, 149, 193, 197, 233, 277, 317, 337, 401, 449, 457, 541, 613, 641, 653, 673, 709, 757, 809, 821, 877, 1009, 1061, 1093, 1117, 1129, 1201, 1289, 1297, 1381, 1481, 1549, 1733, 1873, 1877, 1913, 1933, 1997, 2017, 2053, 2153, 2213, 2221, 2377, 2417, 2437, 2557, 2797
Offset: 1

Views

Author

Torlach Rush, May 03 2019

Keywords

Comments

Replacing the shorter legs with the sum and absolute difference of the shorter legs may result in an integer-sided triangle, but this is not always the case. For example, {5,12,13}->{7,13,17} and {7,13,17} are the sides of a triangle. However, {60,91,109}->{31,109,151}, but {31,109,151} are not the sides of a triangle. If the replacement results in such a triangle, then the triangle is a scalene integer triangle (A070112) with sides of prime length, and a(n) is a term of A070081.
Sequence provides x-value of solutions to the equation 2*x^2 = y^2 + z^2, with x, y and z primes. - Lamine Ngom, Apr 30 2022

Examples

			13 is a term because 12 +  5 = 17 and 12 -  5 =  7.
17 is a term because 15 +  8 = 23 and 15 -  8 =  7.
37 is a term because 35 + 12 = 47 and 35 - 12 = 23.

Crossrefs

Cf. A070081, A070112.
Subset of A008846.
Subset of A307880.
Showing 1-9 of 9 results.

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