A329950 -id:A329950 - cp's OEIS frontend

A345665 Ceiling of circumradius of quadrilateral with consecutive prime sides configured as a cyclic quadrilateral.

4, 6, 7, 9, 11, 13, 16, 19, 22, 25, 28, 30, 33, 36, 40, 43, 46, 49, 52, 55, 58, 62, 66, 70, 73, 75, 77, 81, 86, 91, 95, 99, 102, 106, 110, 113, 117, 121, 124, 129, 132, 135, 138, 142, 147, 153, 158, 162, 165, 167, 171, 175, 179, 184, 188

Offset: 1

Frank M Jackson, Jun 21 2021

nonn

The first cyclic quadrilateral in this sequence with sides (2,3,5,7) is analogous to an obtuse triangle in that the circumcenter does not lie within the bounds of the quadrilateral. Thereafter, the quadrilaterals have circumcenters that lie within the bounds of the quadrilateral.

			a(2)=6 because a cyclic quadrilateral with sides (3,5,7,11) has circumradius = 5.56365...

Eric Weisstein's World of Mathematics, Cyclic Quadrilateral.
Wikipedia, Cyclic quadrilateral.
Wikipedia, Prime triplet.

Cf. A329950, A331676.

lst = {}; Do[{a, b, c, d}={Prime[n], Prime[n+1], Prime[n+2], Prime[n+3]}; s=(a+b+c+d)/2; R=Sqrt[(a*b+c*d)(a*c+b*d)(a*d+b*c)/((s-a)(s-b)(s-c)(s-d))]/4; AppendTo[lst, Ceiling@R], {n, 1, 100}]; lst

The circumradius R of a cyclic quadrilateral with sides a, b, c, d is given by the Parameshvara's circumradius formula R = sqrt((a*b+c*d)*(a*c+b*d)*(a*d+b*c)/((s-a)*(s-b)*(s-c)*(s-d)))/4 where s = (a+b+c+d)/2.

cp's OEIS Frontend

A345665 Ceiling of circumradius of quadrilateral with consecutive prime sides configured as a cyclic quadrilateral.

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