A257547 Even side of integer-sided triangle such that the area is an integer and two sides are twin primes.

4, 20, 452, 2180, 4052, 6500, 27380, 43220, 118100, 130052, 183620, 281252, 357860, 399620, 410420, 455060, 656660, 1134020, 1401140, 1609220, 1630820, 2142452, 4482020, 7258052, 8446052, 8694452, 9618500, 10424180, 11838980, 12370340, 12852452, 13343780

Offset: 1

Author

Michel Lagneau, Apr 29 2015

Keywords

• nonn

Comments

The sides of each triangle are of the form (k^2+2, k^2+4, 2k^2+2)=(p, p+2,q) where p and p+2 primes (see A257049) => the value of the even side is q = 2k^2+2 and the greatest prime divisor of a(n) is equal to the greatest prime divisor of p-1.

The following table gives the first values (n, a(n)=q, A, p, p+2) where A is the integer area.

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

| n | a(n)=q | A | p | p+2 |

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

| 1 | 4 | 6 | 3 | 5 |

| 2 | 20 | 66 | 11 | 13 |

| 3 | 452 | 6810 | 227 | 229 |

| 4 | 2180 | 72006 | 1091 | 1093 |

| 5 | 4052 | 182430 | 2027 | 2029 |

| 6 | 6500 | 370614 | 3251 | 3253 |

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

Note that q = 2*p - 2. - Zak Seidov, May 21 2015

Alternatively, numbers 2k^2 + 2 such that k^2 + 2 and k^2 + 4 are prime. - Charles R Greathouse IV, May 21 2015

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001097, A257049.

Programs

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,2*Prime[c]-2]]],{c,nn}];Union[lst]

PARI

for(k=1,1e4,if(isprime(k^2+2) && isprime(k^2+4), print1(2*k^2+2", "))) \\ Charles R Greathouse IV, May 21 2015