A070126 -id:A070126 - 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 *)

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

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

Reinhard Zumkeller, May 05 2002

nonn

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

Jean-François Alcover, Table of n, a(n) for n = 1..365
Reinhard Zumkeller, Integer-sided triangles

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

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 *)

A070100 Number of integer triangles with perimeter n and prime side lengths which are acute and isosceles.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 0, 2, 0, 1, 1, 2, 0, 2, 1, 2, 0, 2, 0, 2, 0, 1, 1, 2, 0, 3, 0, 2, 0, 1, 0, 3, 0, 1, 1, 2, 0, 2, 1, 3, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 1, 3, 0, 2, 0, 3, 0, 0, 1, 3, 0, 3, 1, 3, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070093, A059169, A070088, A070092, A070095, A070098, A070126.

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

Reinhard Zumkeller, May 05 2002

nonn

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

R. Zumkeller, Integer-sided triangles

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

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.

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

Reinhard Zumkeller, May 05 2002

nonn

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

R. Zumkeller, Integer-sided triangles

Cf. A070098, A070126, A070125, A070115, 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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A070111 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer triangle with prime sides.

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

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A070100 Number of integer triangles with perimeter n and prime side lengths which are acute and isosceles.

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A070117 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles 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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A070124 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute isosceles integer triangle.

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