A330496 - OEIS

%I #16 Jul 27 2021 23:16:45

%S 960,5005,17017,46189,96577,212625,394240,765049,1361920,2027025,

%T 3065857,4385745,6314112,8973909,12780049,17116960,21191625,27428544,

%U 33980800,42600829,56581525,72382464,89835424,107972737,121330189,135745657,167244385,204917929

%N Squared area of quadrilateral with sides prime(n), prime(n+1), prime(n+2), prime(n+3) of odd primes configured as a cyclic quadrilateral. Sequence index starts at n=2 to omit the even prime.

%C If a, b, c, d are consecutive odd primes configured as a cyclic quadrilateral, then Brahmagupta's formula K = sqrt((a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d))/16 means that K^2 will always be an integer. The only cyclic quadrilateral with consecutive prime sides starting with side 2 has a rational squared area of 3003/16.

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

%F 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.

%e a(2)=960 because cyclic quadrilateral with sides 3,5,7,11 has squared area = (3+5+7-11)(3+5-7+11)(3-5+7+11)(-3+5+7+11)/16 = 960.

%t lst = {}; Do[{a, b, c, d} = {Prime[n], Prime[n+1], Prime[n+2], Prime[n+3]}; A2=(a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)/16; AppendTo[lst, A2], {n, 1, 100}]; lst

%Y Cf. A131019, A330096.

%K nonn

%O 2,1

%A _Frank M Jackson_, Dec 16 2019