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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A070126 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute isosceles integer triangle with prime side lengths.

Original entry on oeis.org

3, 6, 9, 16, 22, 34, 35, 43, 46, 63, 84, 109, 124, 159, 170, 189, 201, 234, 286, 297, 350, 352, 382, 410, 450, 478, 479, 515, 527, 597, 629, 688, 708, 811, 817, 868, 900, 981, 1021, 1033, 1105, 1153, 1284, 1386, 1419, 1425
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(5)=22: [A070080(22), A070081(22), A070082(22)]=[3<5=5], A070085(22)=3^2+5^2-5^2=9>0.

Links

Crossrefs

Cf. A070100, A070115, A070111, A070118.

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.

A070134 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

5, 14, 32, 52, 61, 104, 118, 133, 146, 163, 202, 242, 246, 266, 314, 342, 404, 437, 467, 472, 504, 542, 547, 577, 619, 625, 714, 757, 801, 807, 853, 907, 957, 1015, 1022, 1082, 1139, 1145, 1265, 1278, 1335, 1414, 1475
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(5)=61: [A070080(61), A070081(61), A070082(61)]=[5=5<9], A070084(69)=gcd(5,5,9)=1, A070085(61)=5^2+5^2-9^2=25+25-81=-31<0.

Links

Crossrefs

Cf. A070107, A070110, A070127, A070115.

A070135 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle with prime side lengths.

Original entry on oeis.org

5, 14, 133, 163, 472, 542, 714, 907, 1888, 2064, 2267, 2466, 3133, 4240, 4858, 5188, 7563, 8469, 9466, 9958, 11069, 12841, 13471, 14876, 17032, 17794, 20214, 20268, 21125, 21943, 22843, 22933, 23837, 25767, 25797
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(3)=133: [A070080(133), A070081(133), A070082(133)]=[7=7<11], A070085(133)=7^2+7^2-11^2=49+49-121=-23<0.

Links

Crossrefs

Cf. A070108, A070111, A070127, A070115.

A070136 Numbers m such that [A070080(m), A070081(m), A070082(m)] is a right integer triangle.

Original entry on oeis.org

17, 116, 212, 370, 493, 850, 1297, 1599, 1629, 2574, 2778, 3751, 4298, 4370, 5251, 5286, 6476, 9169, 10066, 12398, 12441, 12520, 14414, 16365, 16602, 19831, 21231, 21486, 24060, 26125, 27245, 29230, 33625, 33658
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

Right integer triangles have integer areas: see A070142.

Examples

			116 is a term: [A070080(116), A070081(116), A070082(116)]=[6,8,10], A070085(116)=6^2+8^2-10^2=36+64-100=0.
212 is a term: [A070080(212), A070081(212), A070082(212)]=[5,12,13], A070085(212)=5^2+12^2-13^2=25+144-169=0.

Links

Crossrefs

Cf. A024155, A070137, A070142, A070085.

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^2 + b^2 == c^2] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

A070137 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a right integer triangle with relatively prime side lengths.

Original entry on oeis.org

17, 212, 493, 1297, 2574, 4298, 5251, 14414, 16365, 21231, 26125, 39056, 42597, 55042, 63770, 75052, 91121, 97256, 124355, 164640, 200999, 213083, 253721, 275999, 367997, 384154, 415778, 478343, 511633, 518370, 606417, 665040, 689356, 755435, 846571
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

Right integer triangles have integer areas: see A070143.

Examples

			493 is a term: [A070080(493), A070081(493), A070082(493)]=[8,15,17], A070084(493)=gcd(8,15,17)=1, A070085(493)=8^2+15^2-17^2=64+225-289=0.

Links

Crossrefs

Cf. A070109, A070136.

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_} /; GCD[a, b, c] == 1 && a^2 + b^2 - c^2 == 0] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

Extensions

More terms from Jean-François Alcover, Oct 04 2021

A070119 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an acute integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 6, 7, 11, 12, 15, 16, 19, 22, 23, 27, 28, 33, 35, 39, 40, 43, 45, 46, 47, 51, 53, 55, 58, 60, 63, 64, 65, 70, 72, 73, 81, 83, 88, 90, 92, 94, 95, 98, 103, 106, 107, 108, 109, 114, 119, 121, 124, 132, 134, 135, 136, 140, 142, 148
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(15)=33: [A070080(33), A070081(33), A070082(33)]=[4,5,6], A070084(33)=gcd(4,5,6)=1, A070085(33)=4^2+5^2-6^2=16+25-36=5>0.

Links

Crossrefs

Cf. A070094, A070122, A070125, A070118, A070110.

A070128 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse integer triangle with relatively prime side lengths.

Original entry on oeis.org

5, 8, 13, 14, 20, 21, 25, 29, 30, 32, 36, 37, 41, 42, 44, 49, 52, 56, 57, 59, 61, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 86, 89, 96, 97, 99, 100, 101, 102, 104, 105, 110, 111, 113, 115, 118, 122, 123, 125, 126, 127, 128, 130, 131, 133
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(9)=30: [A070080(30), A070081(30), A070082(30)]=[3,5,7], A070084(30)=gcd(3,5,7)=1, A070085(30)=3^2+5^2-7^2=9+25-49=-15>0.

Links

Crossrefs

Cf. A070102, A070131, A070134, A070110, A070127.

A070120 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle with prime side lengths.

Original entry on oeis.org

3, 6, 9, 16, 22, 34, 35, 43, 46, 63, 84, 109, 124, 159, 170, 189, 201, 234, 240, 286, 297, 350, 352, 382, 410, 450, 478, 479, 515, 527, 544, 597, 629, 688, 708, 799, 811, 817, 868, 900, 911, 981, 1021, 1033, 1105, 1153, 1262
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(6)=34: [A070080(34), A070081(34), A070082(34)]=[5,5,5], A070085(34)=5^2+5^2-5^2=25>0.

Links

Crossrefs

Cf. A070095, A070126, A070123, A070111, A070118.
Previous Showing 11-20 of 27 results. Next

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