A257049 -id:A257049 - cp's OEIS frontend

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

Michel Lagneau, Apr 29 2015

nonn

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

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

Cf. A001097, 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,2*Prime[c]-2]]],{c,nn}];Union[lst]

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

A283222 Integer area of integer-sided triangle such that the sides are of the form p, p+2, 2(p-1), where p, p+2 and (p-1)/2 are prime numbers.

Original entry on oeis.org

66, 6810, 182430, 105470250, 17356640970, 678676246650, 1879504308930, 4491035717130, 10618004862030, 21136679055030, 23751520478010, 27081671511090, 27596192489190, 31721097756750, 115248550935750, 133303609919430, 140838829659930, 182797297112430, 197799116497230

Offset: 1

Michel Lagneau, Mar 03 2017

nonn

Subsequence of 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.
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).
The corresponding primes p are a subsequence of A056899 (primes of the form n^2+2): 11, 227, 2027, 140627, 4223027, 48650627, 95942027, 171479027, ...
We observe that p == 11 mod 72, or p == 11, 83 mod 144. For p>11, p == 27, 227, 627 mod 1000.
An interesting property: the greatest prime divisor of a(n) is equal to p. For instance, the prime divisors of 6810 are {2, 3, 5, 227} => p = 227 is the length of the smallest side of the triangle (227, 229, 452).
The following table gives the first values of A, the sides of the triangles and the primes (p-1)/2.
+-----------+--------+--------+--------+---------+
|         A |      p |    p+2 |  2(p-1)| (p-1)/2 |
+-----------+--------+--------+--------+---------+
|        66 |     11 |     13 |     20 |       5 |
|      6810 |    227 |    229 |    452 |     113 |
|    182430 |   2027 |   2029 |   4052 |    1013 |
| 105470250 | 140627 | 140629 | 281252 |   70313 |
+-----------+--------+--------+--------+---------+

			66 is in the sequence because the area of the triangle (11, 13, 20) is given by Heron's formula with s = 22 and A = sqrt(22(22-11)(22-13)(22-20)) = 66. The numbers 11, 13 and 5 = (11-1)/2 are primes.

Cf. A056899, A085554, A086381, A188158, A253639, A257049.

nn:=100000:
for n from 1 by 2 to nn do:
if isprime(n^2+2) and isprime(n^2+4) and isprime((n^2+1)/2)
then
printf(`%d, `,n*(2*n^2+4)):
else
fi:
od:

nn=10000;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 && PrimeQ[(Prime[c]-1)/2], AppendTo[lst,Sqrt[area2]]]], {c,nn}];Union[lst]

lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isprime((p-1)/2), ca = p; cb = p+2; cc = 2*(p-1); sp = (ca+cb+cc)/2; a2 = sp*(sp-ca)*(sp-cb)*(sp-cc); if (issquare(a2), print1(sqrtint(a2), ", "));););} \\ Michel Marcus, Mar 04 2017

a(n) == 6 mod 30.

cp's OEIS Frontend

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

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