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Search: id:A105548
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| A105548 |
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Continued fraction expansion of prime nested radical A105546. |
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+0
6
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| 2, 9, 1, 1, 1, 7, 3, 5, 4, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 15, 1, 3, 1, 41, 6, 1, 3, 1, 3, 10, 1, 1, 1, 9, 9, 1, 25, 1, 3, 1, 1, 2, 2, 2, 1, 34, 59, 2, 2, 2, 1, 2, 2, 3, 3, 1, 5, 2, 21, 3, 4, 10, 1, 3, 20, 2, 3, 2, 1, 4, 7, 1, 6, 1, 6, 3, 4, 1, 3, 5, 6, 1, 1, 4, 1, 3, 6, 25, 7, 2, 1, 1, 2, 1, 6, 1, 1, 7, 1, 3, 2
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Records are: 9,15,41,59,117,153,599,1663,8212,..., . Robert G. Wilson v: "It would appear superficially that this constant is normal." Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(4 + ... = ~ 1.75793275661800... "It was discovered by T. Vijayaraghavan that the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; A072449] We know the asymptotic limit of primes, and hence that the Prime Nested Radical converges.
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REFERENCES
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Borwein, J. M. and de Barra, G. "Nested Radicals." Amer. Math. Monthly 98, 735-739, 1991.
Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
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LINKS
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Eric Weisstein's World of Mathematics, Nested Radical Constant.
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FORMULA
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Continued fraction expansion of Sqrt(2 + Sqrt(3 + Sqrt(5 + Sqrt(7 + Sqrt(11 + ... + Sqrt(Prime(n))))).
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MATHEMATICA
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Mathematica by Robert G. Wilson v (rgwv(AT)rgwv.com). f[n_] := Block[{k = n, s = 0}, While[k > 0, s = Sqrt[s + k]; k-- ]; s]; ContinuedFraction[ f[100], 101],
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CROSSREFS
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Cf. A000040, A072449, A105546.
Adjacent sequences: A105545 A105546 A105547 this_sequence A105549 A105550 A105551
Sequence in context: A133768 A085333 A128892 this_sequence A094242 A083649 A073009
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KEYWORD
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cofr,easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 14 2005
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