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

Author

Jonathan Vos Post, Jun 09 2007

Keywords

• nonn

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.

Examples

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

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

• 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.

Crossrefs

Cf. A106171, A131020.

Cf. A034963.

Programs

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 *)

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.

Extensions

Edited by R. J. Mathar, Jun 12 2007