A055593 -id:A055593 - cp's OEIS frontend

A070081 Middle side of integer triangles [A070080(n) <= a(n) <= A070082(n)], sorted by perimeter, sides lexicographically ordered.

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

G. C. Greubel, Table of n, a(n) for the first 55 rows, flattened
Reinhard Zumkeller, Integer-sided triangles

Cf. A046129, A055593, A069595, A069594, A069598.

Cf. A070080, A070082, A070083, A070084, A070085, A070086.

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]]&];
triangles[[All, 2]] (* Jean-François Alcover, Jul 09 2017 *)

a(n) = A070083(n) - A070080(n) - A070082(n).

A188158 Area A of the triangles such that A and the sides are integers.

Original entry on oeis.org

6, 12, 24, 30, 36, 42, 48, 54, 60, 66, 72, 84, 90, 96, 108, 114, 120, 126, 132, 144, 150, 156, 168, 180, 192, 198, 204, 210, 216, 234, 240, 252, 264, 270, 288, 294, 300, 306, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 432, 456, 462, 468, 480, 486, 504, 510, 522, 528

Offset: 1

Michel Lagneau, Mar 22 2011

nonn

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2. A given area often corresponds to more than one triangle; for example, a(9) = 60 for the triangles (a,b,c) = (6,25,29), (8,17,15), (13,13,10) and (13,13,24).

If only primitive integer triangles (that is, the lengths of the sides are coprime) are considered, then the possible areas are 6 times the terms in A083875. - T. D. Noe, Mar 23 2011

			a(3) = 24 because the area of the triangle whose sides are 4, 15, 13 is given by sqrt(p(p-4)(p-15)(p-13)) = 24, where p = (4 + 15 + 13)/2 = 16.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangle
Wikipedia, Heronian triangle

Cf. A007237, A009112, A024153, A024365, A051516, A051584, A051585, A055592, A055593, A055594, A055595.

# storage of areas in T(i)
T:=array(1..4000):nn:=100:k:=1:for a from 1
  to nn do: for b from 1 to nn do: for c from 1 to nn do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c):   if x>0 then x1:=abs(x):s:=sqrt(x1) :else fi:if s=floor(s) then T[k]:=s:k:=k+1:else
  fi:od:od:od:
# sort of T(i)
for jj from 1 to k-1 do: ii:=jj:for k1 from  ii+1 to k-1 do:if T[ii]>T[k1] then ii:=k1:else fi:od: m:=T[jj]:T[jj]:=T[ii]:T[ii]:=m:od:liste:=convert(T,set):print(liste):
# second program:
isA188158 := proc(A::integer)
    local Asqr, s,a,b,c ;
    Asqr := A^2 ;
    for s in numtheory[divisors](Asqr) do
        if s^2> A then
        for a from 1 to s-1 do
            if modp(Asqr,s-a) = 0 then
                for b from a to s-1 do
                    c := 2*s-a-b ;
                    if s*(s-a)*(s-b)*(s-c) = Asqr then
                        return true ;
                    end if;
                end do:
            end if;
        end do:
        end if;
    end do:
    false ;
end proc:
for n from 3 to 600 do
    if isA188158(n) then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, May 02 2018

nn = 528; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst] (* T. D. Noe, Mar 23 2011 *)

A055595 Area of triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

6, 12, 12, 24, 48, 30, 60, 54, 24, 84, 48, 36, 60, 120, 108, 66, 42, 96, 84, 126, 60, 108, 192, 90, 150, 84, 168, 120, 36, 204, 240, 210, 210, 60, 120, 216, 132, 300, 96, 336, 72, 192, 144, 240, 480, 294, 84, 252, 360, 432, 114, 156, 180, 210, 420, 120, 210, 420

Offset: 1

Henry Bottomley, May 26 2000

nonn

This is the ordering of triangles used for A316841.

Ray Chandler, Table of n, a(n) for n = 1..10000
A. Dendane, Heron's Formula For Area of a Triangle-Geometry Calculator

The sides are given by A055592, A055593, A055594.
Cf. A046131, A070786, A316841, A316853.
Range of values: A188158.

max = 42; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055595 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 4]](* Jean-François Alcover, Jun 12 2012 *)

a(n) = sqrt(s(n)*(s(n)-A055592(n))*(s(n)-A055593(n))*(s(n)-A055594(n))) where s(n) = (A055592(n)+A055593(n)+A055594(n))/2 i.e. half the perimeter of the triangle

A055592 Longest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

5, 6, 8, 10, 12, 13, 13, 15, 15, 15, 16, 17, 17, 17, 18, 20, 20, 20, 21, 21, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 28, 29, 29, 30, 30, 30, 30, 30, 30, 30, 32, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 37, 37, 39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 40, 40, 41, 41

Offset: 1

Henry Bottomley, May 26 2000

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A055593, A055594, A055595.

max = 41; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055592 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 3]](* Jean-François Alcover, Jun 12 2012 *)

A055594 Shortest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

3, 5, 5, 6, 10, 5, 10, 9, 4, 13, 10, 9, 8, 16, 15, 11, 7, 12, 10, 13, 13, 15, 20, 12, 15, 7, 14, 10, 3, 17, 20, 17, 20, 6, 17, 18, 11, 25, 8, 26, 5, 20, 18, 16, 32, 21, 8, 15, 25, 30, 19, 15, 13, 12, 24, 16, 17, 25, 10, 15, 30, 25, 22, 29, 14, 24, 13, 25, 15, 9, 17, 18, 29, 20, 35

Offset: 1

Henry Bottomley, May 26 2000

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A055592, A055593, A055595.

max = 42; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055594 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 1]](* Jean-François Alcover, Jun 12 2012 *)

A070784 Square of second longest side of triangles with sides whose squares are integers and with positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

2, 4, 5, 4, 5, 8, 4, 8, 10, 10, 6, 5, 8, 9, 12, 15, 5, 8, 10, 13, 13, 5, 8, 13, 16, 17, 10, 16, 8, 9, 10, 10, 13, 13, 16, 17, 17, 17, 18, 20, 20, 12, 15, 8, 13, 13, 16, 17, 17, 20, 20, 25, 10, 10, 16, 18, 20, 20, 20, 26, 24, 14, 9, 13, 16, 20, 20, 20, 25, 25, 29, 12, 24, 30, 30, 10

Offset: 1

Henry Bottomley, May 07 2002

nonn

			a(6)=8 since the triangle with sides sqrt(9), sqrt(8) and sqrt(5) has area 3.

A. Bogomolny, Sam Loyd's Geometric Puzzle

Cf. A055593, A070783, A070785, A070786, A070787.

A336272 Length of longest side of a primitive square Heron triangle, i.e., a triangle with relatively prime integer sides and area the square of a positive integer.

Original entry on oeis.org

17, 26, 120, 370, 392, 567, 680, 697, 847, 1066, 1089, 1183, 1233, 1299, 1371, 1448, 1904, 2009, 2169, 2176, 2281, 2307, 2535, 2600, 2619, 2785, 2845, 2993, 3150, 3370, 3825, 3944, 3983, 4035, 4095, 4290, 4706, 4760, 4879, 4905, 5655, 5811, 5835, 6137, 6375, 6570, 6936, 7202, 7913, 7995

Offset: 1

James R. Buddenhagen, Jul 15 2020

nonn

The triangle [a(23)=2535, 2329, 544] with gcd(2329, 544) = 17 is the first square Heron triangle for which the 3 sides [i, j, k] are not pairwise coprime, i.e., max(gcd(i,j), gcd(i,k), gcd(j,k)) > 1, but gcd(i,j,k) = 1. Are there more square Heron triangles with this property? - Hugo Pfoertner, Jul 18 2020
There are other square Heron triangles with this property, e.g. [a(31)=3825, 2704, 1921] with gcd(1921, 3825) = 17; [a(??)=41460721, 38639097, 17536520] with gcd(38639097, 17536520) = 41; [a(??)=153915025, 139641489, 25224736] with gcd(25224736, 153915025) = 17; and [a(??)=4325561361, 3459908000, 1430190961] with gcd(3459908000, 1430190961) = 73. - James R. Buddenhagen, Jul 20 2020
Terms are given with multiplicity, e.g. if there are two primitive square Heron triangles with equal longest sides, that longest side is listed as a term of the sequence twice (this is very rare). - James R. Buddenhagen, Jul 21 2020

			17 is in the sequence because the triangle with sides [17, 10, 9] has longest side 17 and area 6^2, the square of a positive integer; 26 is in the sequence because the triangle with sides [26, 25, 3] has longest side 26 and has area 6^2, the square of a positive integer.
Triangles with sides [a, b, c] corresponding to the first 8 terms of this sequence are:  [17, 10, 9], [26, 25, 3], [120, 113, 17], [370, 357, 41], [392, 353, 255], [567, 424, 305], [680, 441, 337], [697, 657, 104].

Hugo Pfoertner, Table of n, a(n) for n = 1..79
Sascha Kurz, On the generation of Heronian triangles, arXiv:1401.6150 [math.NT], 11 Jan 2014.
Sascha Kurz, On the generation of Heronian triangles, Serdica Journal of Computing Vol. 2 (2008), Pages 181-196.
Hugo Pfoertner, List of triangle sides, 20000 > i > j > k.

Cf. A046128, A046129, A046130, A046131, A055592, A055593, A055594, A055595.

# find all square Heron triangles whose longest side is between small and big
small:=1: big:=700:
A336272:=[]:triangles:=[]:
areasq16:=(a+b+c)*(a+b-c)*(a-b+c)*(-a+b+c):
# a>=b>=c
for a from small to big do:
  for b from ceil((a+1)/2) to a do:
    for c from a-b+1 to b do:
      if issqr(areasq16) and issqr(sqrt(areasq16)) and igcd(a,b,c)=1 then
        A336272:=[op(A336272),a]:
        triangles:=[op(triangles),[a,b,c]]:
      end if:
    od:
  od:
od: A336272;triangles;

for(a=1,1200,for(b=ceil((a+1)/2),a,for(c=a-b+1,b,if(gcd([a,b,c])==1,if(ispower((a+b+c)*(a+b-c)*(a-b+c)*(b+c-a),4),print1(a,", ")))))) \\ Hugo Pfoertner, Jul 18 2020

a(42)-a(50) from Hugo Pfoertner, Jul 18 2020

cp's OEIS Frontend

A070081 Middle side of integer triangles [A070080(n) <= a(n) <= A070082(n)], sorted by perimeter, sides lexicographically ordered.

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A188158 Area A of the triangles such that A and the sides are integers.

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A055595 Area of triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

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A055592 Longest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

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A055594 Shortest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

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Mathematica

A070784 Square of second longest side of triangles with sides whose squares are integers and with positive integer area, ordered by longest side, then second longest side and finally shortest side.

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A336272 Length of longest side of a primitive square Heron triangle, i.e., a triangle with relatively prime integer sides and area the square of a positive integer.

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