A257547 Even side of integer-sided triangle such that the area is 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

1,1

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

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]

Prog

(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

Crossrefs

Cf. A001097, A257049.

Sequence in context: A054465 A118713 A303630 * A160703 A132511 A144989

Adjacent sequences: A257544 A257545 A257546 * A257548 A257549 A257550

Keyword

nonn

Author

Michel Lagneau, Apr 29 2015

Status

approved