A105546 Decimal expansion of prime nested radical.

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

Offset: 1

Author

Jonathan Vos Post, Apr 12 2005

Keywords

• cons
• nonn

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

Examples

			2.10359749633989726261993964968532544404216228824001387298687284563...
		

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

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.

Programs

Mathematica

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

Formula

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

Extensions

Crossrefs corrected by Jaroslav Krizek, Jan 03 2015