cp's OEIS Frontend

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

%I A338267 #20 Dec 14 2023 18:04:11
%S A338267 0,6,13,38,71,108,159,218,317,436,550,697,817,961,1185,1425,1667,1884,
%T A338267 2134,2377,2635,3009,3438,3931,4351,4645,4888,5200,5778,6548,7485,
%U A338267 7955,8653,9238,10033,10642,11389,12151,12928,13653,14570,15324,16233,16683,17676,19153,20963,22174,22832,23620
%N A338267 a(n) is the nearest integer to the area of a triangle with sides prime(n), prime(n+1), prime(n+2).
%C A338267 It appears that the area is rational only for n=1.
%H A338267 Robert Israel, <a href="/A338267/b338267.txt">Table of n, a(n) for n = 1..10000</a>
%F A338267 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 A338267 a(n) = round(sqrt((3/16)*A330096(n))). - _Hugo Pfoertner_, Oct 19 2020
%e A338267 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 A338267 atr:= proc(p,q,r) local s; s:= (p+q+r)/2; sqrt(s*(s-p)*(s-q)*(s-r)) end proc:
%p A338267 P:= [seq(ithprime(i),i=1..102)]:
%p A338267 seq(round(atr(P[i],P[i+1],P[i+2])),i=1..100);
%t A338267 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 A338267 (Python)
%o A338267 from sympy import prime, integer_nthroot
%o A338267 def A338267(n):
%o A338267     p, q, r = prime(n)**2, prime(n+1)**2, prime(n+2)**2
%o A338267     return (integer_nthroot(4*p*q-(p+q-r)**2,2)[0]+2)//4 # _Chai Wah Wu_, Oct 19 2020
%Y A338267 Cf. A330096, A338262, A338269.
%K A338267 nonn
%O A338267 1,2
%A A338267 _Robert Israel_, Oct 19 2020