A338269 -id:A338269 - 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).

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

Robert Israel, Oct 19 2020

nonn

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

			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.

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

Cf. A330096, A338262, A338269.

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);

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 *)

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

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

A338262 Primes p such that the area of the triangle with sides p and the next two primes achieves a record for closeness to a prime.

Original entry on oeis.org

2, 3, 5, 239, 2521, 12239, 121421, 869657, 23638231, 30656909, 47964149, 48203291, 57273361, 552014783, 754751369, 941234383

Offset: 1

J. M. Bergot and Robert Israel, Oct 19 2020

			a(3)=5 is in the sequence because 5 is a prime, the triangle with sides 5, 7, 11 has area 3*sqrt(299)/4 whose distance to the nearest prime, 13, is approximately 0.0313, and this is less than any distance previously achieved.

Cf. A330096, A338267, A338269.

atr:= proc(p,q,r) local s; s:= (p+q+r)/2; sqrt(s*(s-p)*(s-q)*(s-r)) end proc:
R:= 2,3: p:= 3: q:= 5: r:= 7: count:= 2: dmin:= 7 - atr(3,5,7):
while count < 8 do
p:= q: q:= r: r:= nextprime(r);
a:= atr(p,q,r);
m:= round(a);
if not isprime(m) then next fi;
d:= abs(a-m);
if is(d < dmin) then
  count:= count+1;
  dmin:= d;
  R:= R, p;
fi
od:
R;

lista(nn) = {my(m=p=3, q=5, s, t); print1(2); forprime(r=7, nn, s=sqrt((p-s=(p+q+r)/2)*(q-s)*(s-r)*s); if(m>t=min(s-precprime(s), nextprime(s)-s), print1(", ", p); m=t); p=q; q=r); } \\ Jinyuan Wang, Oct 24 2020

a(10)-a(16) from Jinyuan Wang, Oct 24 2020

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).

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