A330096 -id:A330096 - cp's OEIS frontend

A331676 Ceiling of circumradius of triangle with consecutive prime sides.

5, 8, 7, 9, 10, 12, 15, 17, 20, 22, 24, 26, 28, 32, 34, 37, 39, 41, 44, 46, 49, 53, 56, 58, 60, 62, 64, 68, 72, 77, 79, 82, 85, 89, 91, 94, 97, 101, 103, 107, 109, 112, 114, 117, 123, 128, 131, 133, 135, 138, 141, 145, 149, 152, 155, 158, 160, 162, 166, 171, 176, 180

Offset: 2

Frank M Jackson, Jan 24 2020

nonn

The sequence starts at offset 2 because using the first three primes yields a triangle with sides (2,3,5) that is degenerate with infinite circumradius.

Also the first two triangles in this sequence with sides (3,5,7) and (5,7,11) are obtuse and do not have their circumcentres within the bounds of the triangle. Thereafter, the triangles are acute and their circumcentres lie within the bounds of the triangle.

			a(2)=5 because a triangle with sides 3,5,7 has area = (1/4)*sqrt((3+5+7)(3+5-7)(3-5+7)(-3+5+7)) = 6.495... and circumradius = 3*5*7/(4A) = 4.041...

Harvey P. Dale, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Circumradius.

Cf. A096377, A330096, A331227, A331228.

lst = {}; Do[{a, b, c}={Prime[n], Prime[n+1], Prime[n+2]}; s=(a+b+c)/2; A=Sqrt[s(s-a)(s-b)(s-c)]; R=a*b*c/(4 A); AppendTo[lst, Ceiling@R], {n, 2, 200}]; lst
ccr[{a_,b_,c_}]:=Module[{s=(a+b+c)/2,A},A=Sqrt[s(s-a)(s-b)(s-c)];Ceiling[(a*b*c)/(4A)]]; ccr/@Partition[Prime[Range[2,70]],3,1] (* Harvey P. Dale, Aug 02 2025 *)

Circumradius R of a triangle with sides a, b, c is given by R = a*b*c/(4A) where the area A is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) and where s = (a+b+c)/2.

A338267 a(n) is the nearest integer to the area of a triangle with sides prime(n), prime(n+1), prime(n+2).

Original entry on oeis.org

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

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.

Original entry on oeis.org

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

A330496 Squared area of quadrilateral with sides prime(n), prime(n+1), prime(n+2), prime(n+3) of odd primes configured as a cyclic quadrilateral. Sequence index starts at n=2 to omit the even prime.

Original entry on oeis.org

960, 5005, 17017, 46189, 96577, 212625, 394240, 765049, 1361920, 2027025, 3065857, 4385745, 6314112, 8973909, 12780049, 17116960, 21191625, 27428544, 33980800, 42600829, 56581525, 72382464, 89835424, 107972737, 121330189, 135745657, 167244385, 204917929

Offset: 2

Frank M Jackson, Dec 16 2019

nonn

If a, b, c, d are consecutive odd primes configured as a cyclic quadrilateral, then Brahmagupta's formula K = sqrt((a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d))/16 means that K^2 will always be an integer. The only cyclic quadrilateral with consecutive prime sides starting with side 2 has a rational squared area of 3003/16.

			a(2)=960 because cyclic quadrilateral with sides 3,5,7,11 has squared area = (3+5+7-11)(3+5-7+11)(3-5+7+11)(-3+5+7+11)/16 = 960.

Wikipedia, Cyclic quadrilateral.

Cf. A131019, A330096.

lst = {}; Do[{a, b, c, d} = {Prime[n], Prime[n+1], Prime[n+2], Prime[n+3]}; A2=(a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)/16; AppendTo[lst, A2], {n, 1, 100}]; lst

Area K of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta's formula K = sqrt((s-a)(s-b)(s-c)(s-d)) where s = (a+b+c+d)/2.

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A331676 Ceiling of circumradius of triangle with consecutive prime sides.

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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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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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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.

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Extensions

A330496 Squared area of quadrilateral with sides prime(n), prime(n+1), prime(n+2), prime(n+3) of odd primes configured as a cyclic quadrilateral. Sequence index starts at n=2 to omit the even prime.

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Mathematica

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