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Search: id:A131019
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| A131019 |
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Semiperimeters of quadrilaterals whose sides are 4 consecutive odd primes. |
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+0
4
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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(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 analogue of A106171.
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REFERENCES
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Coolidge, J. L. "A Historically Interesting Formula for the Area of a Quadrilateral." Amer. Math. Monthly 46, 345-347, 1939.
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.
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LINKS
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Eric Weisstein's World of Mathematics, Brahmagupta's Formula.
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FORMULA
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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.
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EXAMPLE
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a(1) = (3 + 5 + 7 + 11)/2 = 13.
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MAPLE
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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:
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CROSSREFS
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Cf. A106171, A131020.
Cf. A034963.
Sequence in context: A108265 A069853 A123909 this_sequence A037158 A066469 A119149
Adjacent sequences: A131016 A131017 A131018 this_sequence A131020 A131021 A131022
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KEYWORD
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nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 09 2007
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EXTENSIONS
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Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 12 2007
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