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

%I A283446 #29 Mar 21 2017 06:33:49
%S A283446 24,96,120,144,168,216,240,264,336,360,384,432,456,480,504,528,576,
%T A283446 600,624,672,720,792,816,840,864,936,960,1008,1056,1080,1176,1200,
%U A283446 1224,1296,1320,1344,1440,1512,1536,1560,1584,1680,1728,1824,1848,1920,1944,2016
%N A283446 Integer areas of triangles with side lengths A001223(m), A001223(m+1) and A001223(m+2) for some m.
%C A283446 A001223(n) = A000040(n+1) - A000040(n) = prime(n+1) - prime(n).
%C A283446 The Mathematica program examines all triangles with n <= 10^8.
%C A283446 The sequence a(n) is a subsequence of A188158, and the lengths of the sides are even.
%C A283446 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.
%C A283446 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)}.
%C A283446 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).
%C A283446 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.
%C A283446 +-----+--------+-----------+-------------+-------------+
%C A283446 |  A  |     m  | A001223(m)| A001223(m+1)| A001223(m+2)|
%C A283446 +-----+--------+-----------+-------------+-------------+
%C A283446 |  24 |    123 |     6     |      8      |     10      |
%C A283446 |  96 |   3935 |    16     |     20      |     12      |
%C A283446 | 120 |   8101 |    10     |     26      |     24      |
%C A283446 | 144 |  13097 |    34     |     18      |     20      |
%C A283446 | 168 |  12226 |    30     |     40      |     14      |
%C A283446 | 216 |   9864 |    24     |     18      |     30      |
%C A283446 | 240 | 102715 |    58     |     50      |     12      |
%C A283446 | 264 |  98259 |    22     |     26      |     40      |
%C A283446 | 336 |  38604 |    30     |     28      |     26      |
%C A283446 +-----+--------+-----------+-------------+-------------+
%e A283446 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.
%e A283446 The set of the others values m > 123 giving the same area A = 24 starts with 127, 192, 269, 304, 417, 420, ...
%t A283446 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]
%Y A283446 Cf. A000040, A001223, A188158, A229746.
%K A283446 nonn
%O A283446 1,1
%A A283446 _Michel Lagneau_, Mar 08 2017
%E A283446 Missing terms 1200 and 1584 from _Giovanni Resta_, Mar 08 2017