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

2,1

Comments

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.

Links

Table of n, a(n) for n=2..63.

Eric Weisstein's World of Mathematics, Circumradius.

Formula

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.

Example

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...

Mathematica

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

Crossrefs

Cf. A096377, A330096, A331227, A331228.

Sequence in context: A357838 A143618 A177056 * A053787 A314569 A314570

Adjacent sequences: A331673 A331674 A331675 * A331677 A331678 A331679

Keyword

nonn

Author

Frank M Jackson, Jan 24 2020

Status

approved