A188158 -id:A188158 - cp's OEIS frontend

A210207 Area A of the non-right triangles such that A, the sides, and the circumradius are integers.

168, 432, 480, 624, 672, 768, 1320, 1512, 1536, 1560, 1680, 1728, 1848, 1920, 2040, 2304, 2376, 2496, 2520, 2688, 2856, 3024, 3072, 3240, 3696, 3720, 3840, 3864, 3888, 4104, 4200, 4320, 4536, 5280, 5376, 5616, 5712, 6000, 6048, 6144, 6240, 6552, 6720, 6912

Offset: 1

Michel Lagneau, Mar 18 2012

nonn

A103251 gives the areas of right triangles with the same property (the area, the sides, and the circumradius are integers). Thus the intersection of this sequence with A103251 will give the areas of 2 families of triangles with the same property: one family of right triangles and one family of non-right triangles.
For example a(3) = A103251(8) = 480 generates two triangles whose sides are
(a,b,c) = (32, 50, 78) = > A = 480, R = 65, and 32^2 + 50^2 is no square;
(a,b,c) = (20, 48, 52) = > A = 480, R = 26, and 20^2 + 48^2  = 52^2 is square.
{a(n) intersection A103251} = {480, 1320, 1536, 1920, 2520, 3024, 3696, 3840, ...}

			168 is in the sequence because, for (a,b,c) = (14,30,40), A = sqrt(42*(42-14)*(42-30)*(42-40)) = 168, and 14^2 + 30^2 is no square.

Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.
Eric W. Weisstein, MathWorld: Circumradius

Cf. A103251, A208984, A188158.

T:=array(1..4000):nn:=400:k:=0:for a from 1
to nn do: for b from a to nn do: for c from b to nn do:  p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): u:=a^2+b^2:if  x>0 then x1:=sqrt(x) : y:=a*b*c/(4*x1):
else fi:if x1=floor(x1) and y = floor(y) and u <> c^2 then k:=k+1:T[k]:=x1:else fi:od:od:od: L := [seq(T[i],i=1..k)]:L1:=convert(T,set):A:=sort(L1, `<`): print(A):

nn=400; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0 < area2 && a^2 != b^2+c^2 && IntegerQ[Sqrt[area2]] && IntegerQ[a*b*c/(4*Sqrt[area2])], AppendTo[lst, Sqrt[area2]]]], {a,nn}, {b,a}, {c,b}]; Union[lst]

Area A = sqrt(s*(s-a)*(s-b)*(s-c)) with s = (a+b+c)/2 (Heron's formula);

Circumradius R = a*b*c/4A.

A231174 Integer areas of integer-sided triangles such that the distance between the incenter and the circumcenter is an integer.

Original entry on oeis.org

48, 192, 432, 768, 1200, 1728, 2352, 3072, 3840, 3888, 4800, 5808, 6000, 6912, 8112, 9408, 10800, 12288, 12960, 13872, 15360, 15552, 17328, 19200, 21168, 23232, 24000, 25392, 26880, 27648, 30000, 30720, 32448, 32928, 34560, 34992, 37632, 40368, 43200, 46128

Offset: 1

Michel Lagneau, Nov 05 2013

nonn

The distance between the incenter and circumcenter is given by d = sqrt(R(R-2r)), where R is the circumradius and r is the inradius, a result known as the Euler triangle formula (see the link below).
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.
The inradius r is given by r = A/s and the circumradius is given by R = abc/4A.
Properties of this sequence:
It appears that the triangles are isosceles.
a(n) = 48*m where the integers m are not squarefree: {m} ={1, 4, 9, 16, 25, 36, 49, 64, 80, 81, 100, 121, 125, 144, 169, 196, 225, 256, 270, 289, ...}, and the areas of the primitive triangles are 48, 3840, 6000, ... The integers m are not squarefree.
The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.
The subsequence of the areas of triangles with inradius, circumradius and distance between the incenter and circumcenter integers is {432, 1728, 3072, 3888, 6000, 6912, ...}.
The following table gives the first values (A, a, b, c, R, r, d) where A is the area of the triangles, a, b, c are the integer sides of the triangles, R is the circumradius, r is the inradius and d is the distance between the incenter and circumcenter:
--------------------------------------------
|    A | a  |  b  |   c |  R   |  r   |  d |
--------------------------------------------
|   48 | 10 |  10 |  16 | 25/3 |  8/3 |  5 |
|  192 | 20 |  20 |  32 | 50/3 | 16/3 | 10 |
|  432 | 30 |  30 |  48 | 25   |  8   | 15 |
|  768 | 40 |  40 |  64 | 100/3| 32/3 | 20 |
| 1200 | 50 |  50 |  80 | 125/3| 40/3 | 25 |
| 1728 | 60 |  60 |  96 |  50  | 16   | 30 |
| 2352 | 70 |  70 | 112 | 175/3| 56/3 | 35 |
| 3072 | 80 |  80 |  96 |  50  | 24   | 10 |
| 3072 | 80 |  80 | 128 | 200/3| 64/3 | 40 |
| 3840 | 80 | 104 | 104 | 169/3| 80/3 | 13 |
| 3888 | 90 |  90 | 144 |  75  | 24   | 45 |
| 4800 |100 | 100 | 160 | 250/3| 80/3 | 50 |
| 5808 |110 | 110 | 176 | 275/3| 88/3 | 55 |
| 6000 |130 | 130 | 240 | 169  | 24   |143 |
| 6912 |120 | 120 | 144 |  75  | 36   | 15 |
| 6912 |120 | 120 | 192 | 100  | 32   | 60 |
..........................................

Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
Eric Weisstein's World of Mathematics, Circumcenter.
Eric Weisstein's World of Mathematics, Incenter.

Cf. A188158.

nn=800;lst={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s (s-a)(s-b)(s-c);If[area2>0&&IntegerQ[Sqrt[area2]]&&IntegerQ[Sqrt[a*b*c/(4*Sqrt[area2])*(a*b*c/(4*Sqrt[area2])-2*Sqrt[area2]/s)]],AppendTo[lst,Sqrt[area2]]]],{a,nn},{b,a},{c,b}];Union[lst]

A289155 Smallest area of triangle with integer sides and area = n times perimeter.

Original entry on oeis.org

24, 84, 192, 336, 540, 756, 1134, 1344, 1710, 2100, 2640, 3000, 4056, 4116, 4680, 5376, 6936, 6804, 8664, 8400, 9240, 10164, 12696, 12000, 13500, 14196, 15390, 16296, 20184, 18720, 23064, 21504, 23232, 24276, 26040, 27000, 32856, 30324

Offset: 1

Zhining Yang, Jun 26 2017

nonn

			For n = 4, a(4)=336 means for the smallest triangle (a,b,c) = (26,28,30), the area is 336, which is 4 times the perimeter 84.

Ray Chandler, Table of n, a(n) for n = 1..5000 (first 300 terms from Zhining Yang)

Cf. A007237, A120572, A188158, A228383.

a(n) is the leading entry in row n of the triangle in A290451.

for(k=1, 50, n=0;A=10^9; d=4*k^2; e=3*d; for(b=1, sqrt(e), for (c=2*k, e/b, if(b*c>d&&c>=b, f = (b + c)*d / (b * c - d); if(f>=c, a=floor(f); if(a==f, n++; s=2*(a+b+c)*k;if(s

a(n) = A120572(2n). - Ray Chandler, Jul 27 2017

A228383 Area A of the triangle such that A, the sides, and the inradius are integers.

Original entry on oeis.org

6, 24, 30, 36, 42, 48, 54, 60, 66, 84, 96, 108, 114, 120, 126, 132, 144, 150, 156, 168, 180, 192, 198, 210, 216, 240, 252, 264, 270, 294, 300, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 432, 456, 462, 468, 480, 486, 504, 510, 522, 528, 540, 546, 570

Offset: 1

Michel Lagneau, Aug 21 2013

nonn

The sequences A208984 and A185210 are subsequences of this sequence. The corresponding inradius r are 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 3, 4, 3, ...
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. The inradius r is given by r = A/s.
a(n) is divisible by 6 and the squares of the form 36k^2 are in the sequence.

			24 is in the sequence because for (a, b, c) = (6, 8, 10) => s =(6 + 8 + 10)/2 = 12; A = sqrt(12*(12-6)*(12-8)*(12-10)) = sqrt(576)= 24; r = A/s = 2.

Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), pp. 32-34.
Eric W. Weisstein, MathWorld: Inradius

Cf. A188158, A120572, A210250, A208984, A185210.

nn = 1000; 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]] && IntegerQ[Sqrt[area2]/s], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

A226453 Integer areas of integer-sided triangles where at least one side is of prime length.

Original entry on oeis.org

6, 12, 24, 30, 36, 42, 60, 66, 72, 84, 90, 114, 120, 126, 132, 156, 180, 204, 210, 216, 234, 240, 252, 264, 270, 288, 300, 306, 330, 336, 360, 390, 396, 420, 456, 462, 504, 510, 522, 528, 546, 570, 624, 630, 660, 684, 690, 714, 720, 756, 780, 798, 840, 864

Offset: 1

Michel Lagneau, Sep 16 2013

nonn

Subset of A188158.
The area of the triangles (a,b,c) are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
There exist triangles where two distinct integer sides are a prime number, for example:
a(n) = 6 with sides (3,4,5);
a(n) = 30 with sides (5,12,13);
a(n) = 66 with sides (11,13,20);
a(n) = 72 with sides (5,29,30);
a(n) = 114 with sides (19,20,37).
The following table gives the first values (A, a, b, c):
**********************
*  A * a  * b  *  c  *
**********************
*  6 *  3 *  4 *  5  *
* 12 *  5 *  5 *  6  *
* 12 *  5 *  5 *  8  *
* 24 *  4 * 13 * 15  *
* 30 *  5 * 12 * 13  *
* 36 *  3 * 25 * 26  *
* 36 *  9 * 10 * 17  *
* 42 *  7 * 15 * 20  *
* 60 *  6 * 25 * 29  *
* 60 *  8 * 15 * 17  *
* 60 * 10 * 13 * 13  *
* 60 * 13 * 13 * 24  *
* 66 * 11 * 13 * 20  *
* 72 *  5 * 29 * 30  *
......................

			114 is in the sequence because the triangle (19, 20, 37) => semiperimeter s = (19+20+37)/2 = 38, and A = sqrt(38*(38-19)*(38-20)*(38-37)) = 114.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A188158.

nn=1000; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); If[0

A229746 Integer areas of integer-sided triangles where two sides are of prime length.

Original entry on oeis.org

6, 12, 30, 60, 66, 72, 114, 120, 180, 210, 240, 330, 336, 360, 396, 420, 456, 660, 756, 780, 840, 900, 984, 1116, 1200, 1248, 1260, 1290, 1320, 1584, 1590, 1680, 1710, 1716, 1770, 1800, 1980, 2100, 2310, 2400, 2460, 2496, 2520, 2604, 2640, 2940, 2970, 3060, 3120

Offset: 1

Michel Lagneau, Sep 28 2013

nonn

Subset of A188158. The length of the third side is an even composite number because the perimeter is always even.
The area of the triangles (a,b,c) are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
The following table gives the first values (A, a, b, c):
***********************
*   A *  a *  b *  c  *
***********************
*   6 *  3 *  4 *  5  *
*  12 *  5 *  5 *  6  *
*  12 *  5 *  5 *  8  *
*  30 *  5 * 12 * 13  *
*  60 * 10 * 13 * 13  *
*  66 * 11 * 13 * 20  *
*  72 *  5 * 29 * 30  *
* 114 * 19 * 20 * 37  *
* 120 * 16 * 17 * 17  *
* 120 * 17 * 17 * 30  *
* 180 * 13 * 30 * 37  *
....................

			114 is in the sequence because the triangle (19, 20, 37) => semiperimeter s = (19+20+37)/2 = 38, and A = sqrt(38*(38-19)*(38-20)*(38-37)) = 114, with 19 and 37 prime numbers.

Eric W. Weisstein, Heron's Formula

Cf. A188158, A229159.

area[a_, b_, c_] := Module[{s = (a + b + c)/2, a2}, a2 = s (s - a) (s - b) (s - c); If[a2 < 0, 0, Sqrt[a2]]]; goodQ[a_, b_, c_] := Module[{ar = area[a, b, c]}, ar > 0 && IntegerQ[ar]]; nn = 80; t = {}; ps = Prime[Range[2, nn]]; mx = 3*ps[[-1]]; Do[If[p <= q && goodQ[p, q, e], aa = area[p, q, e]; If[aa <= mx, AppendTo[t, aa]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* T. D. Noe, Oct 01 2013 *)

Extended by T. D. Noe, Sep 30 2013

Missing term 2970 from Giovanni Resta, Mar 08 2017

A238369 Integer area A of triangles with side lengths in the commutative ring Z[sqrt(2)].

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 81, 82, 84, 85

Offset: 1

Michel Lagneau, Feb 25 2014

nonn

Generalized integer areas triangles in the ring Z[sqrt(2)] = {a + b sqrt(2)| a,b in Z}.
The sequence A188158 is included in this sequence. The numbers 2*A188158(n) are in the sequence because if the integer area of the integer-sided triangle (a, b, c) is A,  the area of the triangle of sides (a*sqrt(2), b*sqrt(2), c*sqrt(2)) is 2*A.
The primitive areas are 1, 3, 7, 9, 10, 15, 17, 19, 21, 25, ... and the numbers 2^p, 3*2^p, 7*2^p, ... are in the sequence. The numbers p^2*a(n) are in the sequence.
According to the limits of the Mathematica program, it is impossible to find integer areas of values 5, 11, 13, 22, 29, 39, 45, 47, 55, 57, 58, 59, 67, 71, 73, 78, 79, 83, 87, ... with sides in the ring Z(sqrt(2)).
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. For the same area, the number of triangles is not unique, for example the area of the triangles (3,4,5), (2,10,6*sqrt(2)),(3,6-sqrt(2),-3+5*sqrt(2)),(3,6+sqrt(2),3+5*sqrt(2)) and (7-4*sqrt(2), 3+7*sqrt(2), 4+7*sqrt(2)) is A = 6.
Geometric property of the triangles in the ring Z[sqrt(2)]
It is possible to obtain integers values (or rational values) for the irradius (and/or) the circumradius of the triangles (see the table below).
The following table gives the first values (A, a, b, c, r, R) where A is the integer area, a,b,c are the sides in Z[sqrt(2)] and r = A/p, R = a*b*c/(4*A) are respectively the values of the irradius and the circumradius.
Notation in the table:
q=sqrt(2) and irrat. = irrational numbers of the form u+v*q.
---------------------------------------------------------
|  A |   a     |      b      |      c |   r   |  R      |
---------------------------------------------------------
|  1 |   q     |      q      |      2 | irrat.|  1      |
|  2 |   1     |      5      |    4*q | irrat.|  irrat. |
|  3 |   6     |      q      |    5*q | irrat.|  5      |
|  4 |   6     |  5-2*q      |  5+2*q |   1/2 | 51/8    |
|  6 |   3     |      4      |      5 |   1   |  5/2    |
|  7 |   2     |    5*q      |    5*q | irrat.|  irrat. |
|  8 |   4     |      4      |    4*q | irrat.|  irrat. |
|  9 |   6     |    3*q      |    3*q | irrat.|  6      |
| 10 | 5*q     |  9-2*q      | -1+3*q | irrat.|  irrat. |
| 12 |   5     |      5      |      6 |   3/2 | 25/8    |
| 14 |   5     |      7      |    4*q | irrat.|  irrat. |
| 15 |  10     | -4+5*q      |  4+5*q | irrat.| 17/3    |
| 16 |   8     |    4*q      |    4*q | irrat.|  4      |
| 17 |  18     | -8+7*q      |  8+7*q | irrat.|  9      |
| 18 |   6     |      6      |    6*q | irrat.|  irrat. |
........................................................

Eric Weisstein's World of Mathematics, Ring

Cf. A188158.

err=1/10^10;nn=40;q=Sqrt[2];lst={};lst1={};Do[If[u+q*v>0,lst=Union[lst,{u+q*v}]],{u,-nn,nn},{v,-nn,nn}];n1=Length[lst];Do[a=Part[lst,i];b=Part[lst,j];c=Part[lst,k];s=(a+b+c)/2;area2=s*(s-a)*(s-b)*(s-c);If[a*b*c !=0&&N[area2]>0&&Abs[N[Sqrt[area2]]-Round[N[Sqrt[area2]]]]

A257049 Integer area of integer-sided triangle such that two sides are twin primes.

Original entry on oeis.org

6, 66, 6810, 72006, 182430, 370614, 3203694, 6353634, 28698786, 33163770, 55637466, 105470250, 151375626, 178631034, 185921166, 217064574, 376267326, 853918566, 1172755854, 1443472134, 1472632266, 2217439890, 6709586934, 13826592870, 17356640970, 18127936590

Offset: 1

Michel Lagneau, Apr 23 2015

nonn

The area of a triangle (a,b,c) is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
Property of the sequence:
We observe that the sides of each triangle are of the form (k^2+2, k^2+4, 2k^2+2) and Heron's formula gives immediately the area k(2k^2+4) => a(n)= 2*A086381(n)*A253639(n).
Let the triangle (a,b,c) = (p,p+2,q) with p prime. Because q = 2t is even, Heron's formula gives the area A = sqrt((p+t+1)(p-t+1)(t-1)(t+1)). Suppose p = t+1, so p-t+1 = 2 and A = 2p*sqrt(t-1). We must have t-1 = k^2 a square, hence p=k^2+2 and q= 2t = 2(k^2+1) = 2p-2.
Consequence: the greatest prime divisor of a(n) is the length of the smallest side of the corresponding triangle if and only if p and p+2 are primes.
This statement is false if we consider a triangle of sides (p,p+2,q) where p and p+2 are composite, or p prime and p+2 composite, or p composite and p+2 prime. Example: the area of the triangle (145, 147, 194) is 10584, but the greatest prime divisor of 10584 = 2^3*3^3*7^2 is 7, and 7 is not the smallest side of the triangle, and 145 is different from 2*194-2.
The following table gives the first values (A, a, b, c) where A is the integer area, a=p, b=p+2 and c are the sides with p prime.
+---------+-------+--------+------+
|       A |  a=p  | b= p+2 |    c |
+---------+-------+--------+------+
|       6 |    3  |    5   |    4 |
|      66 |   11  |   13   |   20 |
|    6810 |  227  |  229   |  452 |
|   72006 | 1091  | 1093   | 2180 |
|  182430 | 2027  | 2029   | 4052 |
|  370614 | 3251  | 3253   | 6500 |
+---------+-------+--------+------+

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A085554, A086381, A188158, A253639.

nn=40000; lst={}; Do[s=(2*Prime[c]-2+Prime[c+1]+Prime[c])/2; If[IntegerQ[s], area2=s (s-2*Prime[c]+2)(s-Prime[c+1])(s-Prime[c]); If[area2>0 && IntegerQ[Sqrt[area2]] && Prime[c+1]==Prime[c]+2, AppendTo[lst, Sqrt[area2]]]], {c, nn}]; Union[lst]

a(n) = 2*A086381(n)*A253639(n). - Zak Seidov, Apr 27 2015

A289156 Largest area of triangles with integer sides and area = n times perimeter.

Original entry on oeis.org

60, 1224, 8436, 34320, 103020, 254040, 546084, 1060896, 1907100, 3224040, 5185620, 8004144, 11934156, 17276280, 24381060, 33652800, 45553404, 60606216, 79399860, 102592080, 130913580, 165171864, 206255076, 255135840, 312875100, 380625960, 459637524, 551258736

Offset: 1

Zhining Yang, Jun 26 2017

			For n = 4, a(4) = 34320 means for the largest triangles (a,b,c) = (66,4225,4289), the area is 34320 which is 4 times the perimeter 8580.

Ray Chandler, Table of n, a(n) for n = 1..5000 (first 100 terms from Zhining Yang)
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A007237, A120573, A188158, A228383, A289155.

Cf. A005900, A007900, A053755.

Table[4 n (2 n^2 + 1) (4 n^2 + 1), {n, 27}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {60, 1224, 8436, 34320, 103020, 254040}, 27] (* or *) Rest@ CoefficientList[Series[12 x (5 + 72 x + 166 x^2 + 72 x^3 + 5 x^4)/(1 - x)^6, {x, 0, 27}], x] (* Michael De Vlieger, Jul 03 2017 *)

Vec(12*x*(5 + 72*x + 166*x^2 + 72*x^3 + 5*x^4)/(1 - x)^6 + O(x^30)) \\ Colin Barker, Jun 28 2017

From Colin Barker, Jun 28 2017: (Start)
G.f.: 12*x*(5 + 72*x + 166*x^2 + 72*x^3 + 5*x^4)/(1 - x)^6.
a(n) = 4*n*(2*n^2 + 1)*(4*n^2 + 1).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6. (End)
a(n) = A120573(2*n). - Ray Chandler, Jul 27 2017
From Elmo R. Oliveira, Sep 01 2025: (Start)
E.g.f.: 4*exp(x)*x*(15 + 138*x + 206*x^2 + 80*x^3 + 8*x^4).
a(n) = 12*A005900(n)*A053755(n) = A053755(n)*A007900(n)/2. (End)

A381336 a(n) is the smallest k > 0 for which a nondegenerate integer-sided triangle (k, k + n, c >= k + n) with an integer area exists.

Original entry on oeis.org

3, 6, 9, 12, 12, 18, 5, 7, 4, 24, 14, 36, 15, 10, 36, 14, 7, 8, 6, 21, 8, 3, 12, 5, 10, 15, 12, 20, 46, 35, 9, 28, 20, 14, 25, 16, 15, 12, 22, 21, 19, 16, 12, 6, 20, 5, 4, 10, 11, 20, 21, 30, 96, 24, 13, 9, 18, 7, 25, 63, 21, 18, 22, 9, 35, 9, 25, 21, 36, 17, 13

Offset: 1

Felix Huber, Mar 16 2025

nonn

Longest sides c are in A381337.

			a(5) = 12 because the nondegenerate integer-sided triangle (12, 12 + 5, 25 >= 12 + 5) has an integer area (90), and there is no smaller k > 0 than 12 that satisfies this condition.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Heron's Formula.

Cf. A103974, A103975, A188158, A379830, A381337.

A381336:=proc(n)
    local k,c,s;
    for k do
        for c from k+n to 2*k+n-1 do
            s:=(n+2*k+c)/2;
            if issqr(s*(s-k)*(s-k-n)*(s-c)) then
                return k
            fi
        od
    od;
end proc;
seq(A381336(n),n=1..71);

cp's OEIS Frontend

A210207 Area A of the non-right triangles such that A, the sides, and the circumradius are integers.

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A231174 Integer areas of integer-sided triangles such that the distance between the incenter and the circumcenter is an integer.

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A289155 Smallest area of triangle with integer sides and area = n times perimeter.

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PARI

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A228383 Area A of the triangle such that A, the sides, and the inradius are integers.

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Mathematica

A226453 Integer areas of integer-sided triangles where at least one side is of prime length.

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Mathematica

A229746 Integer areas of integer-sided triangles where two sides are of prime length.

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A238369 Integer area A of triangles with side lengths in the commutative ring Z[sqrt(2)].

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Mathematica

A257049 Integer area of integer-sided triangle such that two sides are twin primes.

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