cp's OEIS Frontend

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

%I A257049 #50 Nov 08 2016 03:33:02
%S A257049 6,66,6810,72006,182430,370614,3203694,6353634,28698786,33163770,
%T A257049 55637466,105470250,151375626,178631034,185921166,217064574,376267326,
%U A257049 853918566,1172755854,1443472134,1472632266,2217439890,6709586934,13826592870,17356640970,18127936590
%N A257049 Integer area of integer-sided triangle such that two sides are twin primes.
%C 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.
%C A257049 Property of the sequence:
%C A257049 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).
%C A257049 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.
%C A257049 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.
%C A257049 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.
%C A257049 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.
%C A257049 +---------+-------+--------+------+
%C A257049 |       A |  a=p  | b= p+2 |    c |
%C A257049 +---------+-------+--------+------+
%C A257049 |       6 |    3  |    5   |    4 |
%C A257049 |      66 |   11  |   13   |   20 |
%C A257049 |    6810 |  227  |  229   |  452 |
%C A257049 |   72006 | 1091  | 1093   | 2180 |
%C A257049 |  182430 | 2027  | 2029   | 4052 |
%C A257049 |  370614 | 3251  | 3253   | 6500 |
%C A257049 +---------+-------+--------+------+
%H A257049 Zak Seidov, <a href="/A257049/b257049.txt">Table of n, a(n) for n = 1..10000</a>
%F A257049 a(n) = 2*A086381(n)*A253639(n). - _Zak Seidov_, Apr 27 2015
%t A257049 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]
%Y A257049 Cf. A085554, A086381, A188158, A253639.
%K A257049 nonn
%O A257049 1,1
%A A257049 _Michel Lagneau_, Apr 23 2015