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A257049 Integer area of integer-sided triangle such that two sides are twin primes. 3
6, 66, 6810, 72006, 182430, 370614, 3203694, 6353634, 28698786, 33163770, 55637466, 105470250, 151375626, 178631034, 185921166, 217064574, 376267326, 853918566, 1172755854, 1443472134, 1472632266, 2217439890, 6709586934, 13826592870, 17356640970, 18127936590 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

Property of the sequence:

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).

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.

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.

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.

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.

+---------+-------+--------+------+

|       A |  a=p  | b= p+2 |    c |

+---------+-------+--------+------+

|       6 |    3  |    5   |    4 |

|      66 |   11  |   13   |   20 |

|    6810 |  227  |  229   |  452 |

|   72006 | 1091  | 1093   | 2180 |

|  182430 | 2027  | 2029   | 4052 |

|  370614 | 3251  | 3253   | 6500 |

+---------+-------+--------+------+

LINKS

Zak Seidov, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = 2*A086381(n)*A253639(n). - Zak Seidov, Apr 27 2015

MATHEMATICA

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]

CROSSREFS

Cf. A085554, A086381, A188158, A253639.

Sequence in context: A068966 A063039 A082781 * A099147 A073326 A024203

Adjacent sequences:  A257046 A257047 A257048 * A257050 A257051 A257052

KEYWORD

nonn

AUTHOR

Michel Lagneau, Apr 23 2015

STATUS

approved

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Last modified January 25 23:08 EST 2020. Contains 331270 sequences. (Running on oeis4.)