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A105815 Decimal expansion of the semiprime nested radical. 6

%I

%S 2,6,6,3,5,2,5,6,3,4,8,0,6,8,5,6,5,4,4,9,8,9,4,4,6,7,3,2,7,2,1,9,5,5,

%T 1,4,5,9,9,9,2,2,9,8,2,6,8,9,2,7,2,9,3,2,9,1,4,8,3,3,7,0,5,8,6,8,0,2,

%U 3,8,8,4,8,7,9,0,3,9,3,2,9,9,3,5,6,4,3,9,6,0,5,6,8,6,4,2,4,5,5,9,9,1,4,5,3

%N Decimal expansion of the semiprime nested radical.

%C The semiprime nested radical is defined by the infinite recursion: sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n))))). This converges by the criterion of 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; Sloane A072449]. The continued fraction representation of this constant is A105816.

%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 & 229.

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

%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 Jonathan Sondow and Petros 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 Limit_{n -> infinity} sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n))))), where semiprime(n) = A001358(n).

%e 2.66352563480685654498944673272195514599922982689272932914833705868...

%t fQ[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]] == 2; RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Select[ Range[260], fQ[ # ] &]]], 10, 111][[1]] (* _Robert G. Wilson v_, May 31 2005 *)

%Y For other nested radicals, see A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105816, A239349.

%Y Cf. A001358.

%K cons,nonn

%O 1,1

%A _Jonathan Vos Post_, Apr 21 2005

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Last modified September 21 02:29 EDT 2020. Contains 337266 sequences. (Running on oeis4.)