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Search: id:A105546
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| A105546 |
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Decimal expansion of prime nested radical. |
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+0
6
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| 2, 1, 0, 3, 5, 9, 7, 4, 9, 6, 3, 3, 9, 8, 9, 7, 2, 6, 2, 6, 1, 9, 9, 3, 9, 6, 4, 9, 6, 8, 5, 3, 2, 5, 4, 4, 4, 0, 4, 2, 1, 6, 2, 2, 8, 8, 2, 4, 0, 0, 1, 3, 8, 7, 2, 9, 8, 6, 8, 7, 2, 8, 4, 5, 6, 3, 8, 8, 5, 1, 7, 0, 8, 4, 8, 3, 7, 3, 6, 2, 3, 2, 1, 8, 4, 6, 6, 9, 7, 4, 7, 6, 3, 3, 5, 5, 2, 1, 9, 4, 4, 9, 4, 0, 9
(list; cons; graph; listen)
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OFFSET
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1,1
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COMMENT
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A105547 is the continued fraction representation of this prime nested radical. A105548 is the similar semiprime nested radical. A105548 is the Fibonacci nested radical. 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; cf. 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 and 229.
Finch, S. R., Analysis of a Radical Expansion, Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.
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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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Sqrt(2 + Sqrt(3 + Sqrt(5 + Sqrt(7 + Sqrt(11 + ... + Sqrt(Prime(n))))).
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EXAMPLE
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2.10359749633989726261993964968532544404216228824001387298687284563...
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MATHEMATICA
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RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Prime[ Range[ 80]]]], 10, 111][[1]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 31 2005)
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CROSSREFS
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Cf. A000040, A072449.
Sequence in context: A099493 A088523 A035543 this_sequence A059297 A077874 A090683
Adjacent sequences: A105543 A105544 A105545 this_sequence A105547 A105548 A105549
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KEYWORD
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cons,nonn
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AUTHOR
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Jonathan Vos Post (jvospost2(AT)yahoo.com), Apr 12 2005
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