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 1-7 of 7 results.

A070127 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse integer triangle.

Original entry on oeis.org

5, 8, 13, 14, 20, 21, 25, 26, 29, 30, 32, 36, 37, 41, 42, 44, 49, 50, 52, 56, 57, 59, 61, 62, 66, 67, 69, 74, 75, 77, 78, 79, 80, 82, 86, 87, 89, 91, 96, 97, 99, 100, 101, 102, 104, 105, 110, 111, 113, 115, 118, 122, 123, 125, 126, 127
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(10)=30: [A070080(30), A070081(30), A070082(30)]=[3,5,7], A070085(30)=3^2+5^2-7^2=9+25-49=-15<0.

Links

Crossrefs

Cf. A070101, A070129, A070130, A070131, A070132, A070133, A070134, A070135, A070128, A070085.

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^2 + b^2 - c^2 < 0] // Flatten (* Jean-François Alcover, Oct 11 2021 *)

A070111 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer triangle with prime sides.

Original entry on oeis.org

3, 5, 6, 9, 14, 16, 22, 30, 34, 35, 43, 46, 63, 84, 101, 109, 124, 133, 153, 159, 163, 170, 189, 193, 201, 234, 240, 286, 297, 328, 334, 350, 352, 382, 392, 410, 444, 450, 454, 472, 478, 479, 515, 519, 527, 542, 544, 597, 603, 621, 629, 688, 708, 714, 771, 777, 795, 799, 811, 817, 868, 878, 900, 907, 911
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			35 is a term: [A070080(35), A070081(35), A070082(35)]=[2,7,7].

Links

Crossrefs

Cf. A070088, A070114, A070117, A070120, A070123, A070126, A070129, A070132, A070135.

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_} /; AllTrue[{a, b, c}, PrimeQ]] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

Extensions

More terms from Jean-François Alcover, Oct 04 2021

A070115 Numbers m such that [A070080(m), A070081(m), A070082(m)] is an isosceles integer triangle.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 18, 19, 22, 23, 24, 26, 27, 28, 31, 32, 34, 35, 38, 39, 40, 43, 46, 47, 48, 51, 52, 54, 55, 58, 61, 63, 64, 65, 68, 71, 72, 73, 76, 81, 82, 84, 85, 88, 91, 93, 94, 95, 98, 103, 104, 107, 108
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			26 is a term because [A070080(26), A070081(26), A070082(26)] = [4=4<6].

Links

Crossrefs

Cf. A059169, A070116, A070117, A070124, A070125, A070126, A070133, A070134, A070135.

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_, a_, b_} | {a_, b_, b_}] // Flatten (* Jean-François Alcover, Oct 12 2021 *)

A070108 Number of integer triangles with perimeter n and prime side lengths which are obtuse and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(k)<=1 until k = 140, for k = 141 there are A005044(141)=432 integer triangles, a(141)=2 as
[37=37<67]: 37+37+67 = 141 and 2*(37^2)<67^2 and 37, 67 are primes,
[41=41<59]: 41+41+59 = 141 and 2*(41^2)<59^2 and 41, 59 are primes.

Links

Crossrefs

Cf. A070080, A070081, A070082, A070101, A059169, A070088, A070092, A070103, A070106, A070135.

A070117 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with prime side lengths.

Original entry on oeis.org

3, 5, 6, 9, 14, 16, 22, 34, 35, 43, 46, 63, 84, 109, 124, 133, 159, 163, 170, 189, 201, 234, 286, 297, 350, 352, 382, 410, 450, 472, 478, 479, 515, 527, 542, 597, 629, 688, 708, 714, 811, 817, 868, 900, 907, 981, 1021, 1033
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(10)=43: [A070080(43), A070081(43), A070082(43)]=[3,7,7].

Links

Crossrefs

Cf. A070092, A070126, A070135, A070111, A070115.

A070129 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse integer triangle with prime side lengths.

Original entry on oeis.org

5, 14, 30, 101, 133, 153, 163, 193, 328, 334, 392, 444, 454, 472, 519, 542, 603, 621, 714, 771, 777, 795, 878, 907, 1005, 1123, 1135, 1508, 1526, 1538, 1694, 1818, 1848, 1858, 1888, 1999, 2023, 2037, 2064, 2066, 2193
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

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

Links

Crossrefs

Cf. A070103, A070132, A070135, A070111, A070127.

A070133 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an obtuse isosceles integer triangle.

Original entry on oeis.org

5, 14, 26, 32, 52, 61, 82, 91, 104, 118, 133, 146, 163, 182, 202, 219, 242, 246, 266, 291, 314, 342, 347, 372, 404, 432, 437, 467, 472, 504, 542, 547, 577, 582, 619, 625, 663, 709, 714, 751, 757, 801, 807, 853, 858, 907, 913
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(5)=52: [A070080(52), A070081(52), A070082(52)]=[5=5<8], A070085(52)=5^2+5^2-8^2=25+25-64=-14<0.

Links

Crossrefs

Cf. A070106, A070134, A070135, A070115, A070127.
Showing 1-7 of 7 results.

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