A131019 Semiperimeters of quadrilaterals whose sides are 4 consecutive odd primes.

13, 18, 24, 30, 36, 44, 51, 60, 69, 76, 84, 92, 101, 110, 120, 129, 136, 145, 153, 162, 174, 185, 195, 204, 210, 216, 228, 240, 254, 267, 278, 288, 298, 310, 319, 330, 341, 350, 362, 372, 381, 390, 400, 415, 430, 445, 456, 464, 471, 482, 494, 506, 520, 530

Offset

1,1

Comments

(2+3+5+7)/2 = 8.5, not an integer. Hence we restrict to odd primes. The cyclic quadrilaterals whose areas, rounded, are prime are given in A131020. The prime semiperimeters begin: a(1) = 13, a(13) = 101. This arises in the cyclic quadrilateral analog of A106171.

References

Coxeter, H. S. M. and Greitzer, S. L. "Cyclic Quadrangles; Brahmagupta's Formula", Sect. 3.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1967.

Links

Table of n, a(n) for n=1..54.

J. L. Coolidge, A Historically Interesting Formula for the Area of a Quadrilateral, Amer. Math. Monthly 46, 345-347, 1939.

Eric Weisstein's World of Mathematics, Brahmagupta's Formula.

Formula

a(n) = (prime(n) + prime(n+1) + prime(n+2) + prime(n+3))/2 for n>1.

a(n) = (prime(n+1) + prime(n+2) + prime(n+3) + prime(n+4))/2 = A034963(n)/2.

Example

a(1) = (3 + 5 + 7 + 11)/2 = 13.

Maple

A131019 := proc(n) local i ; add( ithprime(n+i), i=1..4)/2 ; end: for n from 1 to 180 do printf("%d, ", A131019(n)) : od:

Mathematica

Plus@@@Partition[Prime[Range[2, 6! ]], 4, 1]/2 (* Vladimir Joseph Stephan Orlovsky, Feb 18 2010 *)

Crossrefs

Cf. A106171, A131020.

Cf. A034963.

Sequence in context: A292003 A257373 A346561 * A317771 A037158 A318080

Adjacent sequences: A131016 A131017 A131018 * A131020 A131021 A131022

Keyword

nonn

Author

Jonathan Vos Post, Jun 09 2007

Extensions

Edited by R. J. Mathar, Jun 12 2007

Status

approved