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A229159 Smallest integer areas of integer-sided triangles where at least one side is of length prime(n). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 |

LINKS

Table of n, a(n) for n=1..48.

MAPLE

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:

CROSSREFS

Cf. A226453.

Sequence in context: A279535 A125510 A117859 * A102901 A014435 A175550

Adjacent sequences:  A229156 A229157 A229158 * A229160 A229161 A229162

KEYWORD

nonn

AUTHOR

Michel Lagneau, Sep 17 2013

STATUS

approved

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Last modified November 17 20:57 EST 2018. Contains 317278 sequences. (Running on oeis4.)