A070082 -id:A070082 - cp's OEIS frontend

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

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

Reinhard Zumkeller, May 05 2002

nonn

Right integer triangles have integer areas: see A070142.

			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.

Jean-François Alcover, Table of n, a(n) for n = 1..137
Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Right Triangle.
Reinhard Zumkeller, Integer-sided triangles

Cf. A024155, A070137, A070142, A070085.

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

Reinhard Zumkeller, May 05 2002

nonn

Right integer triangles have integer areas: see A070143.

			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.

Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Right Triangle.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070109, A070136.

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 *)

More terms from Jean-François Alcover, Oct 04 2021

A070209 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer inradius.

Original entry on oeis.org

17, 116, 212, 269, 368, 370, 493, 561, 587, 659, 850, 1204, 1297, 1582, 1599, 1629, 1920, 1988, 2115, 2352, 2555, 2574, 2774, 2778, 3251, 3473, 3746, 3751, 4286, 4298, 4307, 4313, 4319, 4330, 4370, 4406, 5008, 5251

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(3)=212: [A070080(212), A070081(212), A070082(212)] = [5,12,13], for s = A070083(212)/2 = (5+12+13)/2 = 15: inradius = sqrt((s-5)*(s-12)*(s-13)/s) = sqrt(10*3*2/15) = sqrt(4) = 2; therefore A070200(212)=2. [Corrected by _Rick L. Shepherd_, May 15 2008]

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Eric Weisstein's World of Mathematics, Incircle.
R. Zumkeller, Integer-sided triangles

Cf. A070142, A070201, A051516.

A070113 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a scalene integer triangle with relatively prime side lengths.

Original entry on oeis.org

8, 13, 17, 20, 21, 25, 29, 30, 33, 36, 37, 41, 42, 44, 45, 49, 53, 56, 57, 59, 60, 62, 66, 67, 69, 70, 74, 75, 77, 78, 79, 80, 83, 86, 89, 90, 92, 96, 97, 99, 100, 101, 102, 105, 106, 110, 111, 113, 114, 115, 119, 122, 123, 125, 126

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			36 is a term [A070080(36), A070081(36), A070082(36)]=[3<6<7], A070084(36)=gcd(3,6,7)=1.

Jean-François Alcover, Table of n, a(n) for n = 1..596
Reinhard Zumkeller, Integer-sided triangles

Cf. A005044, A070122, A070131, A070112, A070110.

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

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

Reinhard Zumkeller, May 05 2002

nonn

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

R. Zumkeller, Integer-sided triangles

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

Reinhard Zumkeller, May 05 2002

nonn

			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.

Reinhard Zumkeller, Integer-sided triangles

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

Reinhard Zumkeller, May 05 2002

nonn

			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.

R. Zumkeller, Integer-sided triangles

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

A070149 Areas of integer Heronian triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))].

Original entry on oeis.org

6, 12, 12, 24, 30, 24, 48, 36, 54, 48, 60, 60, 42, 84, 66, 84, 96, 108, 60, 120, 36, 90, 126, 108, 84, 60, 120, 150, 72, 96, 168, 120, 192, 132, 204, 210, 210, 84, 144, 216, 192, 240, 114, 156, 180, 120, 240, 300, 168, 210, 168

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			A070142(2)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore a(2)=A070086(39)=area=4*3=12.

Jean-François Alcover, Table of n, a(n) for n = 1..1265
Eric Weisstein's World of Mathematics, Heronian Triangle.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070142, A070210.

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]] &];
area[{a_, b_, c_}] := With[{p = (a+b+c)/2}, Sqrt[p(p-a)(p-b)(p-c)]];
Select[area /@ triangles, IntegerQ] (* Jean-François Alcover, Oct 12 2021 *)

a(n) = A070086(A070142(n)).

A070200 Inradii of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

Triangles [A070080(A070209(n)), A070081(A070209(n)), A070082(A070209(n))] have integer inradii = a(A070209(k))= A070210(k).

			[A070080(25), A070081(25), A070082(25)] = [3,5,6] and s = A070083(25)/2 = (3+5+6)/2 = 7: a(25) = sqrt((s-3)*(s-5)*(s-6)/7) = sqrt((7-3)*(7-5)*(7-6)/7) = sqrt(4*2*1/7) = sqrt(8/7) = 1.069, rounded = 1.

Eric Weisstein's World of Mathematics, Incircle.
R. Zumkeller, Integer-sided triangles

Cf. A070086.

a(n) = sqrt((s-u)*(s-v)*(s-w)/s), where u=A070080(n), v=A070081(n), w=A070082(n) and s=A070083(n)/2=(u+v+w)/2.

A135622 16*Area^2 of integer triangles [A070080(n),A070081(n),A070082(n)].

Original entry on oeis.org

3, 15, 48, 35, 63, 128, 63, 135, 243, 240, 320, 99, 231, 275, 495, 384, 576, 768, 143, 351, 455, 819, 975, 560, 896, 1008, 1344, 195, 495, 675, 1215, 735, 1575, 1875, 768, 1280, 1536, 2048, 2304, 255, 663, 935, 1683, 1071, 2295, 2499, 2975, 1008, 1728

Offset: 1

Franz Vrabec, Feb 29 2008

			A070080(4)=1, A070081(4)=3, A070082(4)=3, so a(4)=(1+3+3)*(-1+3+3)*(1-3+3)*(1+3-3)=35.

See the formula section for the relationships with A070080, A070081, A070082, A070086.
Cf. A317182 (range of values), A331011 (nonunique values), A331250 (counts triangles by area).
Cf. A316853 (with terms ordered as for A316841), and using this order for other sets of triangles: A046131, A055595, A070786.

a(n)=(u+v+w)*(-u+v+w)*(u-v+w)*(u+v-w), where u=A070080(n), v=A070081(n), w=A070082(n).

A070086(n) = round(sqrt(a(n))/4).

cp's OEIS Frontend

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

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

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A070209 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer inradius.

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A070113 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a scalene integer triangle with relatively prime side lengths.

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

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

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

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A070149 Areas of integer Heronian triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))].

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A070200 Inradii of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

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A135622 16*Area^2 of integer triangles [A070080(n),A070081(n),A070082(n)].