A338267 a(n) is the nearest integer to the area of a triangle with sides prime(n), prime(n+1), prime(n+2).

0, 6, 13, 38, 71, 108, 159, 218, 317, 436, 550, 697, 817, 961, 1185, 1425, 1667, 1884, 2134, 2377, 2635, 3009, 3438, 3931, 4351, 4645, 4888, 5200, 5778, 6548, 7485, 7955, 8653, 9238, 10033, 10642, 11389, 12151, 12928, 13653, 14570, 15324, 16233, 16683, 17676, 19153, 20963, 22174, 22832, 23620

Offset: 1

Author

Robert Israel, Oct 19 2020

Keywords

• nonn

Comments

It appears that the area is rational only for n=1.

Examples

			a(3)=13 because the third, fourth and fifth primes are 5,7,11, the area of a triangle with sides 5, 7, 11 is 3*sqrt(299)/4, and the nearest integer to that is 13.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A330096, A338262, A338269.

Programs

Maple

atr:= proc(p,q,r) local s; s:= (p+q+r)/2; sqrt(s*(s-p)*(s-q)*(s-r)) end proc:
P:= [seq(ithprime(i),i=1..102)]:
seq(round(atr(P[i],P[i+1],P[i+2])),i=1..100);

Mathematica

aTr[{a_,b_,c_}]:=Module[{s=(a+b+c)/2},Round[Sqrt[s(s-a)(s-b)(s-c)]]]; aTr/@Partition[Prime[ Range[ 60]],3,1] (* Harvey P. Dale, Dec 14 2023 *)

Python

from sympy import prime, integer_nthroot
def A338267(n):
    p, q, r = prime(n)**2, prime(n+1)**2, prime(n+2)**2
    return (integer_nthroot(4*p*q-(p+q-r)**2,2)[0]+2)//4 # Chai Wah Wu, Oct 19 2020

Formula

a(n) = round(sqrt(s*(s-prime(n))*(s-prime(n+1))*(s-prime(n+2)))) where s = (prime(n)+prime(n+1)+prime(n+2))/2.

a(n) = round(sqrt((3/16)*A330096(n))). - Hugo Pfoertner, Oct 19 2020