A345665 - OEIS

%I #15 Dec 23 2024 22:56:48

%S 4,6,7,9,11,13,16,19,22,25,28,30,33,36,40,43,46,49,52,55,58,62,66,70,

%T 73,75,77,81,86,91,95,99,102,106,110,113,117,121,124,129,132,135,138,

%U 142,147,153,158,162,165,167,171,175,179,184,188

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

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclicQuadrilateral.html">Cyclic Quadrilateral</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cyclic_quadrilateral">Cyclic quadrilateral</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_triplet">Prime triplet</a>.

%F 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.

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

%t 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

%Y Cf. A329950, A331676.

%K nonn

%O 1,1

%A _Frank M Jackson_, Jun 21 2021