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.

A070110 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 27, 28, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 72, 73, 74, 75, 77
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Comments

A070084(a(k)) = gcd(A070080(a(k)), A070081(a(k)), A070082(a(k))) = 1;
all integer triangles [A070080(a(k)), A070081(a(k)), A070082(a(k))] are mutually nonisomorphic.

Examples

			13 is a term: [A070080(13), A070081(13), A070082(13)]=[2,4,5], A070084(13)=gcd(2,4,5)=1.

Links

Crossrefs

Cf. A051493, A070113, A070116, A070119, A070122, A070125, A070128, A070131, A070134, A070137, A070084.

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

A070118 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 15, 16, 18, 19, 22, 23, 24, 27, 28, 31, 33, 34, 35, 38, 39, 40, 43, 45, 46, 47, 48, 51, 53, 54, 55, 58, 60, 63, 64, 65, 68, 70, 71, 72, 73, 76, 81, 83, 84, 85, 88, 90, 92, 93, 94, 95, 98, 103, 106, 107, 108
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(21)=33: [A070080(33), A070081(33), A070082(33)]=[4,5,6], A070085(33)=4^2+5^2-6^2=16+25-36=5>0.

Links

Crossrefs

Cf. A070093, A070119, A070120, A070121, A070122, A070123, A070124, A070125, A070126.

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^2 + b^2 - c^2 > 0] // Flatten (* 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 *)

A070099 Number of integer triangles with perimeter n and relatively prime side lengths which are acute and isosceles.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 4, 2, 2, 2, 5, 1, 4, 2, 4, 3, 6, 2, 6, 3, 4, 3, 5, 3, 8, 3, 4, 3, 8, 3, 9, 5, 5, 4, 10, 3, 9, 4, 6, 5, 11, 4, 8, 5, 7, 6, 12, 3, 13, 6, 8, 7, 9, 4, 14, 7, 8, 5, 15, 5, 15, 7, 9, 8, 13, 6, 16, 6, 11, 8, 17, 5, 13
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Links

Crossrefs

Cf. A070080, A070081, A070082, A070093, A059169, A051493, A070091, A070094, A070098, A070084, A070125.

A070116 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with relatively prime side lengths.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 11, 12, 14, 15, 16, 19, 22, 23, 27, 28, 32, 35, 39, 40, 43, 46, 47, 51, 52, 55, 58, 61, 63, 64, 65, 72, 73, 81, 88, 94, 95, 98, 103, 104, 107, 108, 109, 118, 121, 124, 133, 135, 136, 140, 146, 150, 151, 159, 163, 166
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(10)=15: [A070080(15), A070081(15), A070082(15)]=[3<4=4], A070084(15)=gcd(3,4,4)=1.

Links

Crossrefs

Cf. A070091, A070125, A070134, A070115, A070110.

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.

A070124 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute isosceles integer triangle.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 15, 16, 18, 19, 22, 23, 24, 27, 28, 31, 34, 35, 38, 39, 40, 43, 46, 47, 48, 51, 54, 55, 58, 63, 64, 65, 68, 71, 72, 73, 76, 81, 84, 85, 88, 93, 94, 95, 98, 103, 107, 108, 109, 112, 117, 120, 121, 124, 129
Offset: 1

Views

Author

Reinhard Zumkeller, May 05 2002

Keywords

Examples

			a(13)=18: [A070080(18), A070081(18), A070082(18)]=[4=4=4], A070085(18)=4^2+4^2-4^2=16>0.

Links

Crossrefs

Cf. A070098, A070126, A070125, A070115, A070118.
Showing 1-7 of 7 results.

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