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.

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

Author

Robert Israel, Oct 19 2020

Keywords

• nonn
• more

Examples

			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.
		

Crossrefs

Cf. A330096, A338262, A338267.

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:= 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;

PARI

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

Extensions

a(13)-a(25) from Jinyuan Wang, Oct 24 2020