A338267 - OEIS

%I #20 Dec 14 2023 18:04:11

%S 0,6,13,38,71,108,159,218,317,436,550,697,817,961,1185,1425,1667,1884,

%T 2134,2377,2635,3009,3438,3931,4351,4645,4888,5200,5778,6548,7485,

%U 7955,8653,9238,10033,10642,11389,12151,12928,13653,14570,15324,16233,16683,17676,19153,20963,22174,22832,23620

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

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

%H Robert Israel, <a href="/A338267/b338267.txt">Table of n, a(n) for n = 1..10000</a>

%F 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.

%F a(n) = round(sqrt((3/16)*A330096(n))). - _Hugo Pfoertner_, Oct 19 2020

%e 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.

%p atr:= proc(p,q,r) local s; s:= (p+q+r)/2; sqrt(s*(s-p)*(s-q)*(s-r)) end proc:

%p P:= [seq(ithprime(i),i=1..102)]:

%p seq(round(atr(P[i],P[i+1],P[i+2])),i=1..100);

%t 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 *)

%o (Python)

%o from sympy import prime, integer_nthroot

%o def A338267(n):

%o     p, q, r = prime(n)**2, prime(n+1)**2, prime(n+2)**2

%o     return (integer_nthroot(4*p*q-(p+q-r)**2,2)[0]+2)//4 # _Chai Wah Wu_, Oct 19 2020

%Y Cf. A330096, A338262, A338269.

%K nonn

%O 1,2

%A _Robert Israel_, Oct 19 2020