A070123 -id:A070123 - cp's OEIS frontend

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

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

Reinhard Zumkeller, May 05 2002

nonn

			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.

Jean-François Alcover, Table of n, a(n) for n = 1..431
Reinhard Zumkeller, Integer-sided triangles

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

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 *)

A070112 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle.

Original entry on oeis.org

8, 13, 17, 20, 21, 25, 29, 30, 33, 36, 37, 41, 42, 44, 45, 49, 50, 53, 56, 57, 59, 60, 62, 66, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 86, 87, 89, 90, 92, 96, 97, 99, 100, 101, 102, 105, 106, 110, 111, 113, 114, 115, 116, 119, 122

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(17)=50: [A070080(50), A070081(50), A070082(50)]=[4<6<8].

Jean-François Alcover, Table of n, a(n) for n = 1..920
Reinhard Zumkeller, Integer-sided triangles

Cf. A005044, A070113, A070114, A070121, A070122, A070123, A070130, A070131, A070132.

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] // Flatten (* Jean-François Alcover, Oct 12 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

Reinhard Zumkeller, May 05 2002

nonn

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

Reinhard Zumkeller, Integer-sided triangles

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

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 *)

More terms from Jean-François Alcover, Oct 04 2021

A070097 Number of integer triangles with perimeter n and prime side lengths which are both acute and scalene.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A005044, A070088, A070090, A070095, A024154, A070123.

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

Reinhard Zumkeller, May 05 2002

nonn

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

R. Zumkeller, Integer-sided triangles

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

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

Reinhard Zumkeller, May 05 2002

nonn

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

R. Zumkeller, Integer-sided triangles

Cf. A070095, A070126, A070123, A070111, A070118.

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

Reinhard Zumkeller, May 05 2002

nonn

			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.

R. Zumkeller, Integer-sided triangles

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

cp's OEIS Frontend

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

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A070112 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle.

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

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A070097 Number of integer triangles with perimeter n and prime side lengths which are both acute and scalene.

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

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

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

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