A338262 -id:A338262 - 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

A338269 Odd primes p such that the area of the triangle with sides p and the next two primes achieves a record for closeness to an integer.

Original entry on oeis.org

3, 5, 103, 149, 337, 491, 1559, 1753, 5009, 12239, 44381, 219097, 2789881, 3137357, 4012297, 4171337, 4217693, 5910397, 6837499, 23800489, 53253617, 994831501, 2894057281, 3415613611, 39349394531

Offset: 1

Robert Israel, Oct 19 2020

			a(3)=103 is in the sequence because 103 is a prime, the triangle with sides 103 and the next two primes 107 and 109 has area sqrt(382278435)/4 whose distance to the nearest integer, 4888, is approximately 0.0145, and this is less than any distance previously achieved.

Cf. A330096, A338262, A338267.

atr:= proc(p,q,r) local s; s:= (p+q+r)/2; sqrt(s*(s-p)*(s-q)*(s-r)) end proc:
p:= 2: q:= 3: r:= 5: count:= 0: R:= NULL: dmin:= infinity:
while count < 10 do
  p:= q; q:= r; r:= nextprime(r);
  a:= atr(p,q,r);
  d:= abs(a - round(a));
  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); forprime(r=7, nn, s=sqrt((p-s=(p+q+r)/2)*(q-s)*(s-r)*s); if(m>t=min(s-floor(s), ceil(s)-s), print1(p, ", "); m=t); p=q; q=r); } \\ Jinyuan Wang, Oct 24 2020

a(13)-a(25) 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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