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A105546 Decimal expansion of prime nested radical. 7
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; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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 log(a_n)/2^n exists." [Clawson, 229; cf. A072449].

We know the asymptotic limit of primes and hence that the Prime Nested Radical converges.

Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + ...))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, ... for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - Jonathan Sondow, Mar 25 2014

REFERENCES

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 and 229.

S. R. Finch, Analysis of a Radical Expansion, Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.

_Herman P. Robinson_, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.

Eric Weisstein's World of Mathematics, Nested Radical Constant.

Wikipedia, Tirukkannapuram Vijayaraghavan

FORMULA

sqrt(2 + sqrt(3 + sqrt(5 + sqrt(7 + sqrt(11 + ... + sqrt(prime(n) + ...)))).

EXAMPLE

2.10359749633989726261993964968532544404216228824001387298687284563...

MATHEMATICA

RealDigits[Fold[Sqrt[#1 + #2] &, 0, Reverse[Prime[Range[ 80]]]], 10, 111][[1]] (* Robert G. Wilson v, May 31 2005 *)

CROSSREFS

Cf. A000040, A072449, A239349.

A105548 is the continued fraction representation of this prime nested radical.

A105815 is the similar semiprime nested radical.

A105817 is the Fibonacci nested radical.

Sequence in context: A145460 A202178 A035543 * A059297 A267222 A077874

Adjacent sequences:  A105543 A105544 A105545 * A105547 A105548 A105549

KEYWORD

cons,nonn

AUTHOR

Jonathan Vos Post, Apr 12 2005

EXTENSIONS

Crossrefs corrected by Jaroslav Krizek, Jan 03 2015

STATUS

approved

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Last modified March 24 06:12 EDT 2017. Contains 283984 sequences.