A330496 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.

960, 5005, 17017, 46189, 96577, 212625, 394240, 765049, 1361920, 2027025, 3065857, 4385745, 6314112, 8973909, 12780049, 17116960, 21191625, 27428544, 33980800, 42600829, 56581525, 72382464, 89835424, 107972737, 121330189, 135745657, 167244385, 204917929

Offset

2,1

Comments

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.

Links

Table of n, a(n) for n=2..29.

Wikipedia, Cyclic quadrilateral.

Formula

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.

Example

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.

Mathematica

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

Crossrefs

Cf. A131019, A330096.

Sequence in context: A348523 A158412 A247723 * A057666 A046788 A206079

Adjacent sequences: A330493 A330494 A330495 * A330497 A330498 A330499

Keyword

nonn

Author

Frank M Jackson, Dec 16 2019

Status

approved