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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. 0
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 (list; graph; refs; listen; history; text; internal format)
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 cylic 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: A179672 A158412 A247723 * A057666 A046788 A206079

Adjacent sequences:  A330493 A330494 A330495 * A330497 A330498 A330499

KEYWORD

nonn

AUTHOR

Frank M Jackson, Dec 16 2019

STATUS

approved

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Last modified March 5 11:27 EST 2021. Contains 341823 sequences. (Running on oeis4.)