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.

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

Offset: 1

Author

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

Keywords

• nonn
• more

Examples

			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.
		

Crossrefs

Cf. A330096, A338267, A338269.

Programs

Maple

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;

PARI

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

Extensions

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