A338267 -id:A338267 - cp's OEIS frontend

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

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

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

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.

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