A131020 For all cyclic quadrilaterals with four consecutive primes as sides that have an area that is prime after rounding, the sequence gives the first of these four consecutive primes.

3, 5, 13, 17, 61, 67, 97, 139, 157, 163, 173, 223, 271, 349, 353, 419, 479, 503, 541, 691, 701, 743, 877, 941, 1013, 1049, 1051, 1097, 1123, 1229, 1231, 1249, 1297, 1301, 1423, 1453, 1493, 1531, 1559, 1607, 1621, 1697, 1811, 1901, 1999, 2017, 2027, 2053, 2087

Offset

1,1

Comments

The semiperimeters of cyclic quadrilaterals with four consecutive odd prime sides are given in A131019. 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..49.

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(k) for some k such that, where S = semiperimeter = (prime(k) + prime(k+1) + prime(k+2) + prime(k+3))/2 is an element of A131019 and rounded area = round(sqrt((S-prime(k))*(S-prime(k+1))*(S-prime(k+2))*(S-prime(k+3)))) is prime.

Example

a(5) = 61 because (61 + 67 + 71 + 73)/2 = 136 and sqrt((136 - 61)*(136 - 67)*(136 - 71)*(136 - 73)) = 4603.43622 and round(4603.43622) = 4603 is prime.

Maple

Digits := 80 : isA131020 := proc(p) local p2, p3, p4, s, area; if isprime(p) then p2 := nextprime(p) ; p3 := nextprime(p2) ; p4 := nextprime(p3) ; s := (p+p2+p3+p4)/2 ; area := round(sqrt((s-p)*(s-p2)*(s-p3)*(s-p4))) ; RETURN(isprime(area)) ; else false ; fi ; end: for n from 1 to 380 do if isA131020(ithprime(n)) then printf("%d, ", ithprime(n)) ; fi ; od;

Crossrefs

Cf. A000040, A106171, A131019.

Sequence in context: A284241 A284305 A306388 * A283398 A051900 A084706

Adjacent sequences: A131017 A131018 A131019 * A131021 A131022 A131023

Keyword

easy,nonn

Author

Jonathan Vos Post, Jun 09 2007

Extensions

Edited by R. J. Mathar, Jun 12 2007

Status

approved