A096377 -id:A096377 - cp's OEIS frontend

A330096 a(n) = (16/3) * squared area of a triangle with sides prime(n), prime(n+1), and prime(n+2).

0, 225, 897, 7905, 27265, 61985, 134225, 253825, 537425, 1015105, 1613745, 2587585, 3562545, 4925921, 7485985, 10837585, 14816945, 18923905, 24289265, 30135105, 37040465, 48281105, 63033425, 82430705, 100968945, 115087105, 127426145, 144189185

Offset: 1

Hugo Pfoertner, Dec 01 2019

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A096377.

squaredarea(a,b,c)={my(s=(a+b+c)/2);s*(s-a)*(s-b)*(s-c)};
for(n=1,28,print1((16/3)*squaredarea(prime(n),prime(n+1),prime(n+2)),", "))

A331676 Ceiling of circumradius of triangle with consecutive prime sides.

Original entry on oeis.org

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.

A329950 Floor of area of quadrilateral with consecutive prime sides configured as a cyclic quadrilateral.

Original entry on oeis.org

13, 30, 70, 130, 214, 310, 461, 627, 874, 1167, 1423, 1750, 2094, 2512, 2995, 3574, 4137, 4603, 5237, 5829, 6526, 7522, 8507, 9478, 10390, 11014, 11650, 12932, 14314, 16053, 17799, 19278, 20698, 22159, 23994, 25403, 27190, 29033, 30595, 32718, 34558, 36255, 38014

Offset: 1

Frank M Jackson, Nov 25 2019

nonn

Because it is possible to generate triangles using three consecutive odd primes (see A096377) any four consecutive primes will form a quadrilateral. If such quadrilaterals are configured to be cyclic they will have maximal area. This sequence comprises the integer part of these maximal areas.

Proof: Given 4 consecutive odd primes a, b, c, d with a < b < c < d we only have to prove that a+b+c > d for a quadrilateral to exist. However we know that 3 consecutive odd primes will form a triangle hence a+b > c and a+b+c > 2c and by Bertrand's postulate there exists a prime d such that c < d < 2c so a+b+c > d. By induction this can be extended such that n consecutive primes will always form an n-gon.

			a(1)=13 because the area of the cyclic quadrilateral with sides 2,3,5,7 is (1/4)*sqrt((2+3+5-7)(2+3-5+7)(2-3+5+7)(-2+3+5+7)) = 13.699...

Wikipedia, Cyclic quadrilateral, Bertrand's postulate

Cf. A096377.

lst = {}; Do[{a, b, c, d} = {Prime[n], Prime[n+1], Prime[n+2], Prime[n+3]}; s=(a+b+c+d)/2; A=Sqrt[(s-a)(s-b)(s-c)(s-d)]; AppendTo[lst, IntegerPart@A], {n, 1, 200}]; lst

The 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.

cp's OEIS Frontend

A330096 a(n) = (16/3) * squared area of a triangle with sides prime(n), prime(n+1), and prime(n+2).

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A331676 Ceiling of circumradius of triangle with consecutive prime sides.

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A329950 Floor of area of quadrilateral with consecutive prime sides configured as a cyclic quadrilateral.

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Mathematica

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