A226453 -id:A226453 - cp's OEIS frontend

A229159 Smallest integer areas of integer-sided triangles where at least one side is of length prime(n).

0, 6, 6, 42, 66, 24, 36, 114, 966, 60, 930, 114, 126, 1290, 4230, 90, 1770, 330, 2814, 14910, 216, 4740, 1494, 420, 420, 510, 6180, 4494, 840, 570, 8382, 11790, 630, 9174, 210, 4530, 840, 2934, 45090, 3276, 22554, 1260, 24066, 336, 1386, 16716, 26586, 52182

Offset: 1

Michel Lagneau, Sep 17 2013

nonn

Conjecture: for all prime p > 2 there exists an integer-sided triangle with integer area where at least one side is of length p.
There exist triangles of integer area and integer side lengths having two sides whose lengths are distinct prime numbers; for example, (3,4,5), (11,13,20), (19, 20,37), (43,61,68), (59,68,109), (11,60,61), (79,241, 312), (41,50,89), (26,73,97), ... corresponding to the areas 6, 66, 114, 1290, 1770, 330, 4740, 420, 420, ...
Observation: there exist some integer-area, integer-sided triangles with two prime sides such that the perimeter equals 4 times the smaller prime. For example:
    (3,   4,   5) =>  12 = 4*3;
   (11,  13,  20) =>  44 = 4*11;
   (19,  20,  37) =>  76 = 4*19;
   (43,  61,  68) => 172 = 4*43;
   (59,  68, 109) => 236 = 4*59;
  (131, 181, 212) => 524 = 4*131;
  (139, 157, 260) => 556 = 4*139;
  (179, 260, 277) => 716 = 4*179.
The first 25 values (prime(n), smallest area, a, b, c) are:
+---------+-------+-----+-----+-----+
| prime(n)|  Area |  a  |  b  |  c  |
+---------+-------+-----+-----+-----+
|     2   |     0 |   0 |   0 |   0 |
|     3   |     6 |   3 |   4 |   5 |
|     5   |     6 |   3 |   4 |   5 |
|     7   |    42 |   7 |  15 |  20 |
|    11   |    66 |  11 |  13 |  20 |
|    13   |    24 |   4 |  13 |  15 |
|    17   |    36 |   9 |  10 |  17 |
|    19   |   114 |  19 |  20 |  37 |
|    23   |   966 |  23 | 140 | 159 |
|    29   |    60 |   6 |  25 |  29 |
|    31   |   930 |  31 |  68 |  87 |
|    37   |   114 |  19 |  20 |  37 |
|    41   |   126 |  15 |  28 |  41 |
|    43   |  1290 |  43 |  61 |  68 |
|    47   |  4230 |  47 | 425 | 468 |
|    53   |    90 |   4 |  51 |  53 |
|    59   |  1770 |  59 |  68 | 109 |
|    61   |   330 |  11 |  60 |  61 |
|    67   |  2814 |  67 |  85 | 116 |
|    71   | 14910 |  71 | 447 | 476 |
|    73   |   216 |   9 |  73 |  80 |
|    79   |  4740 |  79 | 241 | 312 |
|    83   |  1494 |  83 |  85 | 164 |
|    89   |   420 |  41 |  50 |  89 |
|    97   |   420 |  26 |  73 |  97 |

Cf. A226453.

with(numtheory):nn:=500: for m from 2 to 40 do: q:=ithprime(m):ii:=0:for a from 1
  to nn while(ii=0) do: for b from a to nn while(ii=0) do: for c from b to nn while(ii=0) do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x0:= sqrt(x):else fi:if (x0=floor(x0) and a=q) or (x0=floor(x0) and b=q) or (x0=floor(x0) and c=q)then ii:=1: printf ( "%d %d %d %d %d \n",q,x0,a,b,c):
:else fi:od:od:od:od:

A256579 Integer areas of integer-sided triangles where at least one of the three altitudes is of prime length.

Original entry on oeis.org

6, 12, 30, 60, 84, 168, 330, 546, 660, 1092, 1224, 1710, 2448, 3036, 3420, 6072, 6090, 7440, 12180, 12654, 14880, 17220, 19866, 25308, 25944, 34440, 37206, 39732, 51330, 51888, 56730, 74412, 75174, 89460, 97236, 102660, 113460, 123240, 142926, 150348, 176220

Offset: 1

Michel Lagneau, Apr 02 2015

nonn

Subset of A226453.
The corresponding primes are: 3, 3, 5, 5, 7, 7, 11, 13, 11, 13, 17, 19, 17, 23, 19, 23, 29, 31, 29, 37, 31, 41, 43, 37, 47, 41, 53, 43, 59, 47, 61, 53, 67, 71, 73, 59, 61, 79, 83, 67, 89, ...
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 altitudes of a triangle with sides length a, b, c and area A have length given by Ha = 2*A/a, Hb = 2*A/b, Hc = 2*A/c.
Properties of this sequence:
- The sequence is infinite (see the formula below);
- The prime altitude of a triangle is the greatest prime divisor of a(n) (the proof is easy if we observe the formula);
- There exists two subsets of numbers included into a(n):
Case (i): A subset with right triangles (a,b,c) where a^2 + b^2 = c^2 with area a1(n) = {6, 30, 84, 330, 546, 1224, ...}. The lengths of the prime altitudes are Ha or Hb = a = p. The sides are of the form (p, q, q+1) with p = sqrt(2*q+1) => the sides are equal to (p, (p^2-1)/2, (p^2+1)/2) and a(n) = (p^3-p)/4.
Case (ii): A subset with isosceles triangles formed by two right triangles of the sequence. So, the areas are a2(n) = {12, 60, 168, 660, 1092, 2448, ...} = 2*a1(n). The sides are of the form (a, a, 2*(a-1)) = ((p^2+1)/2, (p^2+1)/2, p^2-1) and Ha = sqrt(2*a-1) = p, a2(n) = 2*a1(n) = (p^3-p)/2.
We did not find a class of non-isosceles and non-right triangles (a, b, c) whose three altitudes include one of prime length.
The following table gives the first values (A, a, b, c, Ha, Hb, Hc) where A is the integer area, a, b, c are the sides and Ha <= Hb <= Hc are the altitudes.
+------+-----+-----+-----+----------+----------+---------+
|   A  |  a  |  b  |  c  |    Ha    |   Hb     |   Hc    |
+------+-----+-----+-----+----------+----------+---------+
|    6 |   3 |   4 |   5 |   12/5   |    3     |    4    |
|   12 |   5 |   5 |   8 |    3     |   24/5   |   24/5  |
|   30 |   5 |  12 |  13 |    5     |   60/13  |   12    |
|   60 |  13 |  13 |  24 |    5     |  120/13  |  120/13 |
|   84 |   7 |  24 |  25 |  168/25  |    7     |   24    |
|  168 |  25 |  25 |  48 |    7     |  336/25  |  336/25 |
|  330 |  11 |  60 |  61 |  660/61  |   11     |   60    |
|  546 |  13 |  84 |  85 | 1092/85  |   13     |   84    |
|  660 |  61 |  61 | 120 |   11     | 1320/61  | 1320/61 |
| 1092 |  85 |  85 | 168 |   13     | 2184/85  | 2184/85 |
| 1224 |  17 | 144 | 145 | 2448/145 |   17     |  144    |
| 1710 |  19 | 180 | 181 | 3420/181 |   19     |  180    |
| 2448 | 145 | 145 | 288 | 4896/145 | 4896/145 |   17    |
+------+-----+-----+-----+----------+----------+---------+

Eric Weisstein's World of Mathematics, Altitude
Eric Weisstein's World of Mathematics, Isosceles Triangle
Eric Weisstein's World of Mathematics, Right Triangle

Cf. A188158, A210643, A210645, A226453.

# program using the formula
lst:={}:for n from 2 to 50 do:p:=ithprime(n):p1:=(p^3-p)/4:p2:=(p^3-p)/2:lst:=lst union {p1} union {p2}:od:print(lst):

nn = 300; lst = {}; Do[s = (a + b + c)/2; area2 = s (s - a) (s - b) (s - c); If[area2>0 && IntegerQ[Sqrt[area2]]&&(PrimeQ[(2*Sqrt[area2])/a]|| PrimeQ[(2*Sqrt[area2])/b]||PrimeQ[(2*Sqrt[area2])/c]), AppendTo[lst, Sqrt[area2]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

a(n) = (prime(n)^3 - prime(n))/4 for the right triangles;

a(n) = (prime(n)^3 - prime(n))/2 for the isosceles triangles.

cp's OEIS Frontend

A229159 Smallest integer areas of integer-sided triangles where at least one side is of length prime(n).

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