A188158 -id:A188158 - cp's OEIS frontend

A256695 Integer areas A of integer-sided triangles (a, b, c) such that the area of the triangle (a+b, a+c, b+c) is also an integer.

12, 48, 108, 192, 300, 432, 588, 768, 972, 1008, 1200, 1452, 1728, 2028, 2352, 2448, 2520, 2700, 2772, 3060, 3072, 3468, 3888, 4032, 4332, 4800, 5292, 5808, 6348, 6912, 7500, 8112, 8748, 9072, 9408, 9792, 10080, 10092, 10800, 11088, 11532, 11628, 12240, 12288

Offset: 1

Michel Lagneau, Apr 08 2015

nonn

The areas of the primitive triangles are 12, 2520, 2772, 3060, 4032, 5808, 9072, 11088, 11628, 17136, 24948, 41580, ...
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 sequence A135453 (numbers of the form 12*n^2) is included in the sequence because a(1) = 12 is a primitive triangle of the subsequence k^2*a(1), k=1,2,3,...
The following table gives the first values (A, A', a, b, c) where A is the integer area of the triangle (a, b, c), A' is the integer area of the triangle (a+b, a+c, b+c).
+-------+--------+------+------+------+
|    A  |    A'  |   a  |   b  |   c  |
+-------+--------+------+------+------+
|   12  |    60  |   5  |   5  |   8  |
|   48  |   240  |  10  |  10  |  16  |
|  108  |   540  |  15  |  15  |  24  |
|  192  |   960  |  20  |  20  |  32  |
|  300  |  1500  |  25  |  25  |  40  |
|  432  |  2160  |  30  |  30  |  48  |
|  588  |  2940  |  35  |  35  |  56  |
|  768  |  3840  |  40  |  40  |  64  |
|  972  |  4860  |  45  |  45  |  72  |
+-------+--------+------+------+------+
We find a majority of isosceles triangles, but there is a subsequence of non-isosceles triangles with areas 2520, 3060, 10080, 11088, ...
+--------+----------+-------+-------+-------+
|    A   |     A'   |   a   |   b   |   c   |
+--------+----------+-------+-------+-------+
|   2520 |   18270  |   29  |  174  |  175  |
|   3060 |   33150  |   39  |  221  |  250  |
|  10080 |   73080  |   58  |  348  |  350  |
|  11088 |   64350  |  150  |  169  |  275  |
|  12240 |   12240  |   78  |  442  |  500  |
|  17136 |   92820  |  168  |  221  |  325  |
|  41580 |  183150  |  250  |  333  |  407  |
+--------+----------+-------+-------+-------+

			a(1) = 12 because, for (a,b,c) = (5, 5, 8) => s = (5+5+8)/2 = 9, and
A = sqrt(9(9-5)(9-5)(9-8)) = sqrt(144) = 12 and the triangle (5+5, 5+8, 5+8) = (10, 13, 13)=> s1 = (10+13+13)/2 = 18, and A1 = sqrt(18(18-10)(18-13)(18-13)) = sqrt(3600) = 60 is an integer.

Cf. A135453, A188158.

nn=1000;lst={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s(s-a)(s-b)(s-c);u=a+b;v=a+c;w=b+c;s1=(u+v+w)/2;area3=s1(s1-u)(s1-v)(s1-w);If[area2>0&&area3>0&&IntegerQ[Sqrt[area2]]&&IntegerQ[Sqrt[area3]], AppendTo[lst,Sqrt[area2]]]],{a,nn},{b,a},{c,b}];Union[lst]

A283222 Integer area of integer-sided triangle such that the sides are of the form p, p+2, 2(p-1), where p, p+2 and (p-1)/2 are prime numbers.

Original entry on oeis.org

66, 6810, 182430, 105470250, 17356640970, 678676246650, 1879504308930, 4491035717130, 10618004862030, 21136679055030, 23751520478010, 27081671511090, 27596192489190, 31721097756750, 115248550935750, 133303609919430, 140838829659930, 182797297112430, 197799116497230

Offset: 1

Michel Lagneau, Mar 03 2017

nonn

Subsequence of A257049.
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.
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).
The corresponding primes p are a subsequence of A056899 (primes of the form n^2+2): 11, 227, 2027, 140627, 4223027, 48650627, 95942027, 171479027, ...
We observe that p == 11 mod 72, or p == 11, 83 mod 144. For p>11, p == 27, 227, 627 mod 1000.
An interesting property: the greatest prime divisor of a(n) is equal to p. For instance, the prime divisors of 6810 are {2, 3, 5, 227} => p = 227 is the length of the smallest side of the triangle (227, 229, 452).
The following table gives the first values of A, the sides of the triangles and the primes (p-1)/2.
+-----------+--------+--------+--------+---------+
|         A |      p |    p+2 |  2(p-1)| (p-1)/2 |
+-----------+--------+--------+--------+---------+
|        66 |     11 |     13 |     20 |       5 |
|      6810 |    227 |    229 |    452 |     113 |
|    182430 |   2027 |   2029 |   4052 |    1013 |
| 105470250 | 140627 | 140629 | 281252 |   70313 |
+-----------+--------+--------+--------+---------+

			66 is in the sequence because the area of the triangle (11, 13, 20) is given by Heron's formula with s = 22 and A = sqrt(22(22-11)(22-13)(22-20)) = 66. The numbers 11, 13 and 5 = (11-1)/2 are primes.

Cf. A056899, A085554, A086381, A188158, A253639, A257049.

nn:=100000:
for n from 1 by 2 to nn do:
if isprime(n^2+2) and isprime(n^2+4) and isprime((n^2+1)/2)
then
printf(`%d, `,n*(2*n^2+4)):
else
fi:
od:

nn=10000;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 && PrimeQ[(Prime[c]-1)/2], AppendTo[lst,Sqrt[area2]]]], {c,nn}];Union[lst]

lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isprime((p-1)/2), ca = p; cb = p+2; cc = 2*(p-1); sp = (ca+cb+cc)/2; a2 = sp*(sp-ca)*(sp-cb)*(sp-cc); if (issquare(a2), print1(sqrtint(a2), ", "));););} \\ Michel Marcus, Mar 04 2017

a(n) == 6 mod 30.

A283446 Integer areas of triangles with side lengths A001223(m), A001223(m+1) and A001223(m+2) for some m.

Original entry on oeis.org

24, 96, 120, 144, 168, 216, 240, 264, 336, 360, 384, 432, 456, 480, 504, 528, 576, 600, 624, 672, 720, 792, 816, 840, 864, 936, 960, 1008, 1056, 1080, 1176, 1200, 1224, 1296, 1320, 1344, 1440, 1512, 1536, 1560, 1584, 1680, 1728, 1824, 1848, 1920, 1944, 2016

Offset: 1

Michel Lagneau, Mar 08 2017

nonn

A001223(n) = A000040(n+1) - A000040(n) = prime(n+1) - prime(n).
The Mathematica program examines all triangles with n <= 10^8.
The sequence a(n) is a subsequence of A188158, and the lengths of the sides are even.
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(n) == 0 mod 24 => {b(n)} = {a(n)/24} = {1, 4, 5, 6, 7, 9, 10, 11, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 33, 34, 35, 36, 39, 40, 42, 44, 45, 49, 51, 54, 55, 56, 60, 63, 64, 65, 70, 72, ...}. It seems that the primes > 19 are not in {b(n)}.
For the same area, the number of distinct triangles is not always unique; for example, the area 336 can be obtained with triangle (30, 28, 26) starting from prime 461983 and also from triangle (34, 20, 42) starting from prime 2473663 (Giovanni Resta, Mar 08 2017).
The following table gives the first values (A, m, sides of the triangles) where A is the area of the triangles and m is the smallest value generating A.
+-----+--------+-----------+-------------+-------------+
|  A  |     m  | A001223(m)| A001223(m+1)| A001223(m+2)|
+-----+--------+-----------+-------------+-------------+
|  24 |    123 |     6     |      8      |     10      |
|  96 |   3935 |    16     |     20      |     12      |
| 120 |   8101 |    10     |     26      |     24      |
| 144 |  13097 |    34     |     18      |     20      |
| 168 |  12226 |    30     |     40      |     14      |
| 216 |   9864 |    24     |     18      |     30      |
| 240 | 102715 |    58     |     50      |     12      |
| 264 |  98259 |    22     |     26      |     40      |
| 336 |  38604 |    30     |     28      |     26      |
+-----+--------+-----------+-------------+-------------+

			24 is in the sequence because, for the smallest value m = 123, we obtain the triangle of sides (A001223(123), A001223(124), A001223(125)) = (6, 8, 10) and the area is given by Heron's formula with s = 12 and A = sqrt(12(12-6)(12-8)(12-10)) = 24.
The set of the others values m > 123 giving the same area A = 24 starts with 127, 192, 269, 304, 417, 420, ...

Cf. A000040, A001223, A188158, A229746.

nn=10^5;lst={};Do[u=Prime[a+1]-Prime[a];v=Prime[a+2]-Prime[a+1]; w=Prime[a+3]-Prime[a+2];s=(u+v+w)/2;If[IntegerQ[s],area2=s (s-u)(s-v)(s-w);If[area2>0&&IntegerQ[Sqrt[area2]],AppendTo[lst,Sqrt[area2]]]],{a,nn}];Union[lst]

Missing terms 1200 and 1584 from Giovanni Resta, Mar 08 2017

A329536 Integer areas of integer-sided triangles where the lengths of two of the sides are cubes.

Original entry on oeis.org

480, 4200, 5148, 7500, 30720, 65520, 268800, 329472, 349920, 480000, 960960, 1684980, 1713660, 1884960, 1966080, 2413320, 2419560, 3061800, 3752892, 4193280, 5467500, 7500000, 8168160, 10022520, 11166960, 17203200, 17915040, 18462300, 21086208, 22394880, 28964040

Offset: 1

Michel Lagneau, Nov 16 2019

nonn

Subset of A188158.
The area of the triangle (a,b,c) are given by Heron's formula, A = sqrt(s(s-a)(s-b)(s-c)), where the side lengths are a, b, c and semiperimeter s = (a+b+c)/2.
The areas of the nonprimitive triangles of sides (a*k^3, b*k^3, c*k^3), k = 1,2,... are in the sequence with the value A*k^6.
There may be multiple triangles with the same area (see the table of examples below).

			The following table gives the initial values of (A, a, b, c):
+--------+------+-------+-------+
|     A  |    a |     b |    c  |
+--------+------+-------+-------+
|    480 |    8 |   123 |   125 |
|   4200 |   70 |   125 |   125 |
|   4200 |  125 |   125 |   240 |
|   5148 |   88 |   125 |   125 |
|   5148 |  125 |   125 |   234 |
|   7500 |  125 |   125 |   150 |
|   7500 |  125 |   125 |   200 |
|  30720 |   64 |   984 |  1000 |
|  65520 |  125 |  2088 |  2197 |
| 268800 |  560 |  1000 |  1000 |
| 268800 | 1000 |  1000 |  1920 |
| 329472 |  704 |  1000 |  1000 |
| 329472 | 1000 |  1000 |  1872 |
| 349920 |  216 |  3321 |  3375 |
.................................

Cf. A188158, A232461.

nn=600;lst={};Do[s=(a^3+b^3+c)/2;If[IntegerQ[s],area2=s (s-a^3)(s-b^3) (s-c);If[0

A351178 Integral area of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.

Original entry on oeis.org

6, 84, 126, 156, 210, 456, 546, 570, 1116, 1170, 1176, 1554, 2046, 2220, 2394, 3096, 3216, 3294, 3354, 3924, 4740, 5124, 6006, 6180, 6510, 7326, 7446, 8760, 9030, 9264, 9906, 10374, 10920, 11466, 12684, 13104, 15210, 16170, 16296, 16716, 17556, 18060, 18090, 18354, 22134, 22860, 23550

Offset: 1

Bernard Schott, Feb 04 2022

nonn

Middle side b is necessarily even, and the two other sides are odd, so all the areas are even numbers.

If b is the middle side with b even >= 4, if k odd = b-a = c-b with 1 <= k <= b/2 - 1, if gcd(b,k) = 1, then, we have area S = sqrt(3*b^2*(b^2-4*k^2))/4.

			a(1) = 6 corresponds to the Pythagorean triple (3, 4, 5), this is the unique right integer-sided triangle in this sequence.
a(2) = 84 for triple (13, 14, 15) (see MacNeill link).
a(3) = 126 for triple (15, 28, 41) (see Penguin reference, entry 126).
a(4) = 156 for triple (15, 26, 37) (see MacNeill link).

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 126, page 122.

John MacNeill, 13, 14, 15 and 15, 26, 37, Mathematical Spectrum, Vol. 21, No. 3, 1989, pp. 83-84.

Cf. A336750, A336755, A336750, A336756, A336757.

Subsequence of A188158.

lista(nn) = {my(list = List()); for (b = 3, nn, for (a = b-floor((b-1)/2), b-1, my(c = 2*b - a); if (gcd([a,b,c]) == 1, my(p = (a+b+c)/2); if (issquare(x=p*(p-a)*(p-b)*(p-c)), listput(list, sqrtint(x)));););); vecsort(Vec(list));} \\ Michel Marcus, Feb 05 2022

Missing terms added by Michel Marcus, Feb 05 2022

A377418 a(n) is the smallest integer k whose set of divisors contains exactly n triples (x,y,z) of distinct divisors considered as integer-sided triangles with integer areas, or 0 if no such k exists.

Original entry on oeis.org

60, 120, 240, 360, 960, 720, 3480, 1440, 1680, 2880, 6600, 2520, 4200, 10440, 5460, 6240, 4680, 5040, 20400, 7800, 18360, 17160, 26520, 10080, 47040, 9360, 15120, 10920, 55080, 20160, 15600, 16380, 34320, 33600, 18720, 27300, 165240, 53040, 37800, 25200, 21840

Offset: 1

Michel Lagneau, Oct 27 2024

nonn

We observe that this sequence is a subsequence of A169823: a(n) == 0 (mod 60).

The area A of a triangle whose sides have lengths x, y, and z is given by Heron's formula: A = sqrt(s*(s-x)*(s-y)*(s-z)), where s = (x+y+z)/2.

			a(3) = 240 because there are 3 triples of divisors (3, 4, 5), (6, 8, 10) and (12, 16, 20) with integer areas 36, 576, 9216 respectively (Pythagorean triples). The first triple is primitive.
a(9)=1680 because there are 9 triples of divisors (3,4,5), (6,8,10), (7,15,20), (12,16,20), (14,30,40), (21,28,35), (28,60,80), (42,56,70), (84,112,140) with 5 Pythagorean triples : (3,4,5), (6,8,10), (21,28,35), (42,56,70), (84,112,70). The other 4 triangles are arbitrary.

Cf. A169823, A188158, A334382.

with(numtheory):
for n from 1 to 41 do:
ii:=0:
for m from 4 to 10^7 while(ii=0) do:it:=0:
 d:=divisors(m):n0:=nops(d):
  for i from 2 to n0-2 do:
   for j from i+1 to n0 do:
     for k from j+1 to n0 do:
       x:=d[i]:y:=d[j]:z:=d[k]:s:=(x+y+z)/2:A:=s*(s-x)*(s-y)*(s-z):
       if A>0 and sqrt(A)=floor(sqrt(A)) then it:=it+1:else
        fi:
       od:
    od:
   od:
    if it=n then printf(`%d %d \n`,it,m):ii:=1:
     else fi:
  od:
od:

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A256695 Integer areas A of integer-sided triangles (a, b, c) such that the area of the triangle (a+b, a+c, b+c) is also an integer.

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A283222 Integer area of integer-sided triangle such that the sides are of the form p, p+2, 2(p-1), where p, p+2 and (p-1)/2 are prime numbers.

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A283446 Integer areas of triangles with side lengths A001223(m), A001223(m+1) and A001223(m+2) for some m.

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A329536 Integer areas of integer-sided triangles where the lengths of two of the sides are cubes.

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A351178 Integral area of primitive integer-sided triangles whose sides a < b < c are in arithmetic progression.

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A377418 a(n) is the smallest integer k whose set of divisors contains exactly n triples (x,y,z) of distinct divisors considered as integer-sided triangles with integer areas, or 0 if no such k exists.

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