A105546 - OEIS

%I #48 Oct 01 2017 03:04:47

%S 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,

%T 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,

%U 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

%N Decimal expansion of prime nested radical.

%C sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ...)))) = 1.75793275661800...

%C "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].

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

%C 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

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

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

%H Chai Wah Wu, <a href="/A105546/b105546.txt">Table of n, a(n) for n = 1..10000</a>

%H Jonathan M. Borwein and G. de Barra, <a href="http://www.jstor.org/stable/2324426">Nested Radicals</a>, Amer. Math. Monthly 98, 735-739, 1991.

%H Herman P. Robinson, <a href="/A257574/a257574.pdf">The CSR Function</a>, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

%H J. Sondow and P. Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NestedRadicalConstant.html">Nested Radical Constant.</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tirukkannapuram_Vijayaraghavan">Tirukkannapuram Vijayaraghavan</a>

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

%e 2.10359749633989726261993964968532544404216228824001387298687284563...

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

%Y Cf. A000040, A072449, A239349.

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

%Y A105815 is the similar semiprime nested radical.

%Y A105817 is the Fibonacci nested radical.

%K cons,nonn

%O 1,1

%A _Jonathan Vos Post_, Apr 12 2005

%E Crossrefs corrected by _Jaroslav Krizek_, Jan 03 2015