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A345665
Ceiling of circumradius of quadrilateral with consecutive prime sides configured as a cyclic quadrilateral.
2
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
1,1
COMMENTS
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.
LINKS
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral.
Wikipedia, Prime triplet.
FORMULA
The circumradius R of a cyclic quadrilateral with sides a, b, c, d is given by the Parameshvara's circumradius formula R = sqrt((ab+cd)(ac+bd)(ad+bc)/((s-a)(s-b)(s-c)(s-d)) where s = (a+b+c+d)/2.
EXAMPLE
a(2)=6 because a cyclic quadrilateral with sides (3,5,7,11) has circumradius = 5.56365...
MATHEMATICA
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
CROSSREFS
Sequence in context: A193627 A256792 A367186 * A343177 A085817 A177688
KEYWORD
nonn
AUTHOR
Frank M Jackson, Jun 21 2021
STATUS
approved