A070082 -id:A070082 - cp's OEIS frontend

A070144 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle with integer area.

17, 116, 212, 252, 368, 370, 493, 561, 587, 659, 839, 850, 1156, 1186, 1196, 1297, 1582, 1599, 1629, 1912, 1920, 2115, 2352, 2555, 2574, 2713, 2774, 2778, 3251, 3473, 3728, 3746, 3751, 4286, 4298, 4307, 4313, 4319

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(2)=116: [A070080(116), A070081(116), A070082(116)] = [6<8<10], area^2 = s*(s-6)*(s-8)*(s-10) with s=A070083(116)/2=(6+8+10)/2=12, area^2=12*6*4*2=64*9 is an integer square, therefore A070086(116)=area=8*3=24.

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

Cf. A070142, A070145.

A070146 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle with integer area.

Original entry on oeis.org

39, 269, 375, 587, 862, 972, 1196, 1955, 1988, 2352, 2555, 2796, 3818, 4319, 4406, 5378, 6522, 6808, 6880, 6890, 6921, 7234, 7360, 8193, 9159, 9207, 10272, 14545, 15004, 15061, 15101, 15216, 15237, 15943, 16502

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(1)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6]: A070085(39)=5^2+5^2-6^2=14>0 and 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 A070086(39)=area=4*3=12.

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

Cf. A070142, A070136, A070147, A070140.

A070147 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an obtuse integer triangle with integer area.

Original entry on oeis.org

52, 252, 368, 372, 561, 659, 839, 957, 1156, 1186, 1204, 1582, 1912, 1920, 1971, 2115, 2713, 2774, 2790, 3251, 3473, 3728, 3746, 4286, 4307, 4313, 4330, 5008, 5272, 5374, 6369, 6389, 6432, 6776, 6881, 7223, 7310, 7341

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(1)=52: [A070080(52), A070081(52), A070082(52)] = [5,5,8]: A070085(52)=5^2+5^2-8^2=-14<0 and area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(52)/2=(5+5+8)/2=9, area^2=9*4*4*1=16*9 is an integer square, therefore A070086(52)=area=4*3=12.

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

Cf. A070142, A070136, A070146, A070141.

A070148 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer Heronian triangle having triangular area.

Original entry on oeis.org

17, 368, 659, 972, 1156, 1599, 1971, 2555, 2574, 3746, 3818, 4298, 4330, 5374, 14325, 14414, 15004, 15943, 16451, 19475, 19615, 24013, 24051, 33950, 63593, 71630, 75052, 79286, 79670, 79921, 84183, 90187, 93290

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			17 is a term: [A070080(17), A070081(17), A070082(17)] = [3,4,5]: A070086(52)=6.

Jean-François Alcover, Table of n, a(n) for n = 1..89
Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Right Triangle.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070142, A000217.

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)]];
Position[triangles, {a_, b_, c_} /; IntegerQ[area[{a, b, c}]] && IntegerQ[Sqrt[1 + 8 area[{a, b, c}]]]] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

A070210 Inradii of integer triangles [A070080(A070209(n)), A070081(A070209(n)), A070082(A070209(n))].

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 2, 4, 3, 4, 4, 3, 2, 4, 5, 3, 6, 4, 6, 6, 6, 4, 6, 3, 4, 3, 6, 4, 5, 4, 3, 6, 5, 7, 8, 6, 4, 6, 8, 7, 8, 9, 3, 9, 5, 6, 9, 8, 10, 6, 6, 6, 9, 8, 4, 8, 9, 7, 10, 6, 10, 12, 6, 12, 12, 5, 3, 7, 8, 10, 4, 9, 10, 11, 6, 12, 3, 6, 9, 12, 12, 7, 8

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070200(A070209(n)).

			A070209(3)=212: [A070080(212), A070081(212), A070082(212)] = [5,12,13], let s = A070083(212)/2 = (5+12+13)/2 = 15 then inradius = sqrt((s-5)*(s-5)*(s-6)/s) = sqrt(10*3*2/15) = sqrt(4) = 2; a(3) = A070200(212) = 2.

Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.
Eric Weisstein's World of Mathematics, Incircle.
R. Zumkeller, Integer-sided triangles

Cf. A070149, A070201.

A316841 Three-column table read by rows giving integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 1, 3, 3, 2, 3, 3, 3, 4, 3, 2, 4, 3, 3, 4, 4, 1, 4, 4, 2, 4, 4, 3, 4, 4, 4, 5, 3, 3, 5, 4, 2, 5, 4, 3, 5, 4, 4, 5, 5, 1, 5, 5, 2, 5, 5, 3, 5, 5, 4, 5, 5, 5, 6, 4, 3, 6, 4, 4, 6, 5, 2, 6, 5, 3, 6, 5, 4, 6, 5, 5, 6, 6, 1, 6, 6, 2, 6, 6, 3, 6, 6, 4, 6, 6, 5, 6, 6, 6, 7, 4, 4, 7, 5, 3, 7, 5, 4, 7, 5, 5, 7, 6, 2, 7, 6, 3, 7, 6, 4, 7, 6, 5

Offset: 1

N. J. A. Sloane, Jul 23 2018, following a suggestion from Donald S. McDonald

			Table begins (imprimitive triples are labeled i):
[1,1,1],
[2,2,1],
[2,2,2],i
[3,2,2],
[3,3,1],
[3,3,2],
[3,3,3],i
[4,3,2],
[4,3,3],
[4,4,1],
[4,4,2],i
[4,4,3],
[4,4,4],i
[5,3,3],
...

Lars Blomberg, Table of n, a(n) for n = 1..9000

There are A002620(k+1) rows that begin with k.
The three columns are A316843, A316844, A316845.
A316849 is a compressed version.
See A316842 for primitive triples.
See A316851 and A316853 & A317182 for perimeter and area.
Other related sequences: A051493, A070080, A070081, A070082, A070110.

for(i=1,6, for(j=1,i, for(k=1,j, if(j+k>i, print1(i,", ",j,", ",k,", "))))) \\ Hugo Pfoertner, Jan 25 2020

A070093 Number of acute integer triangles with perimeter n.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 2, 4, 3, 5, 4, 5, 5, 5, 6, 6, 6, 7, 7, 9, 8, 10, 9, 10, 10, 11, 12, 12, 12, 14, 13, 16, 14, 17, 16, 17, 18, 18, 20, 20, 20, 22, 22, 24, 23, 25, 26, 26, 27, 28, 30, 30, 29, 32, 31, 35, 33, 36, 36, 38, 39, 40, 40

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

An integer triangle [A070080(k) <= A070081(k) <= A070082(k)] is acute iff A070085(k) > 0.

			For n=9 there are A005044(9)=3 integer triangles: [1,4,4], [2,3,4] and [3,3,3]; two of them are acute, as 2^2+3^2<16=4^2, therefore a(9)=2.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Acute Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070094, A070095, A070118.

Cf. A005044, A024155, A070101.

Table[Sum[Sum[(1 - Sign[Floor[(n - i - k)^2/(i^2 + k^2)]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 12 2019 *)

a(n) = A005044(n) - A070101(n) - A024155(n);
a(n) = A042154(n) + A070098(n).
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1-sign(floor((n-i-k)^2/(i^2+k^2)))) * sign(floor((i+k)/(n-i-k+1))). - Wesley Ivan Hurt, May 12 2019

A070083 Perimeters of integer triangles, sorted by perimeter, sides lexicographically ordered.

Original entry on oeis.org

3, 5, 6, 7, 7, 8, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

A005044(p) is the number of all integer triangles having perimeter p.

R. Zumkeller, Integer-sided triangles

maxPer = 19; maxSide = Floor[(maxPer-1)/2]; order[{a_, b_, c_}] := (a+b+c)*maxPer^3 + a*maxPer^2 + b*maxPer + c; triangles = Reap[Do[If[ a+b+c <= maxPer && c-b < a < c+b && b-a < c < b+a && c-a < b < c+a, Sow[{a, b, c}]], {a, 1, maxSide}, {b, a, maxSide}, {c, b, maxSide}]][[2, 1]]; Total /@ Sort[triangles, order[#1] < order[#2] &] (* Jean-François Alcover, Jun 12 2012 *)
maxPer = m = 22; 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]]&]; Total /@ triangles (* Jean-François Alcover, Jul 09 2017 *)

a(n) = A070080(n) + A070081(n) + A070082(n).

A070088 Number of integer-sided triangles with perimeter n and prime sides.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 0, 1, 0, 1, 1, 2, 0, 3, 1, 3, 0, 2, 0, 2, 0, 3, 1, 3, 0, 5, 1, 5, 0, 4, 0, 3, 0, 5, 1, 5, 0, 4, 0, 4, 0, 2, 0, 3, 0, 5, 1, 3, 0, 6, 1, 8, 0, 5, 0, 5, 0, 4, 0, 3, 0, 5, 1, 6, 0, 6, 0, 4, 0, 7, 1, 7, 0, 9, 1, 10, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			For n=15 there are A005044(15)=7 integer triangles: [1,7,7], [2,6,7], [3,5,7], [3,6,6], [4,4,7], [4,5,6] and [5,5,5]: two of them consist of primes, therefore a(15)=2.

R. Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A070090, A070092, A070095, A070097, A070100, A070103, A070105, A070108, A070111.

triangleQ[sides_] := With[{s = Total[sides]/2}, AllTrue[sides, # < s&]];
a[n_] := Select[IntegerPartitions[n, {3}, Select[Range[Ceiling[n/2]], PrimeQ]], triangleQ] // Length; Array[a, 90] (* Jean-François Alcover, Jul 09 2017 *)
Table[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[n - i - k] - PrimePi[n - i - k - 1]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 13 2019 *)

a(n) = A070090(n) + A070092(n) = A070095(n) + A070103(n).

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * c(i) * c(k) * c(n-i-k), where c = A010051. - Wesley Ivan Hurt, May 13 2019

A070101 Number of obtuse integer triangles with perimeter n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 2, 3, 2, 3, 3, 5, 3, 7, 4, 8, 5, 9, 7, 10, 8, 11, 9, 14, 11, 16, 12, 18, 14, 19, 17, 21, 18, 23, 21, 27, 22, 30, 24, 32, 27, 34, 30, 37, 33, 40, 35, 44, 37, 47, 40, 50, 44, 53, 49, 56, 52, 60, 55, 64, 57, 68

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

An integer triangle [A070080(k) <= A070081(k) <= A070082(k)] is obtuse iff A070085(k) < 0.

			For n=14 there are A005044(14)=4 integer triangles: [2,6,6], [3,5,6], [4,4,6] and [4,5,5]; two of them are obtuse, as 3^2+5^2<36=6^2 and 4^2+4^2<36=6^2, therefore a(14)=2.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Obtuse Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070102, A070103, A070127.

Cf. A005044, A024155, A070093.

Table[Sum[Sum[(1 - Sign[Floor[(i^2 + k^2)/(n - i - k)^2]]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 12 2019 *)

a(n) = A005044(n) - A070093(n) - A024155(n).
a(n) = A024156(n) + A070106(n).
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)}
(1-sign(floor((i^2 + k^2)/(n-i-k)^2))) * sign(floor((i+k)/(n-i-k+1))). - Wesley Ivan Hurt, May 12 2019

cp's OEIS Frontend

A070144 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle with integer area.

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A070146 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle with integer area.

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A070147 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an obtuse integer triangle with integer area.

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A070148 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer Heronian triangle having triangular area.

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Mathematica

A070210 Inradii of integer triangles [A070080(A070209(n)), A070081(A070209(n)), A070082(A070209(n))].

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A316841 Three-column table read by rows giving integer sides of proper triangles (i,j,k) with i >= j >= k >= 1, j+k > i.

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PARI

A070093 Number of acute integer triangles with perimeter n.

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A070083 Perimeters of integer triangles, sorted by perimeter, sides lexicographically ordered.

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A070088 Number of integer-sided triangles with perimeter n and prime sides.

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Mathematica

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A070101 Number of obtuse integer triangles with perimeter n.

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