A338262 - OEIS

%I #27 Oct 24 2020 04:14:30

%S 2,3,5,239,2521,12239,121421,869657,23638231,30656909,47964149,

%T 48203291,57273361,552014783,754751369,941234383

%N 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.

%e 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.

%p atr:= proc(p,q,r) local s; s:= (p+q+r)/2; sqrt(s*(s-p)*(s-q)*(s-r)) end proc:

%p R:= 2,3: p:= 3: q:= 5: r:= 7: count:= 2: dmin:= 7 - atr(3,5,7):

%p while count < 8 do

%p p:= q: q:= r: r:= nextprime(r);

%p a:= atr(p,q,r);

%p m:= round(a);

%p if not isprime(m) then next fi;

%p d:= abs(a-m);

%p if is(d < dmin) then

%p   count:= count+1;

%p   dmin:= d;

%p   R:= R, p;

%p fi

%p od:

%p R;

%o (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

%Y Cf. A330096, A338267, A338269.

%K nonn,more

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Oct 19 2020

%E a(10)-a(16) from _Jinyuan Wang_, Oct 24 2020