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 A257547 Even side of integer-sided triangle such that the area is integer and two sides are twin primes. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 9 05:05 EDT 2024. Contains 375027 sequences. (Running on oeis4.)