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A105816 Continued fraction expansion of the semiprime nested radical (A105815).

%I A105816 #30 Feb 16 2025 08:32:57
%S A105816 2,1,1,1,34,1,2,2,2,1,1,5,1,1,1,1,1,1,1,9,7,1,9,1,5,1,5,1,2,7,2,2,3,5,
%T A105816 2,1,10,8,2,3,1,1,1,12,1,1,5,4,4,2,1,1,2,2,4,13,2,2,12,3,11,15,2,2,2,
%U A105816 23,8,1,1,3,1,2,8,19,1,5,2,7,4,1,82,22,1,1,1,2,1,1,9,1,1,1,15,8,12,2,11,1,15
%N A105816 Continued fraction expansion of the semiprime nested radical (A105815).
%C A105816 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].
%C A105816 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 A105816 Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 and 229.
%D A105816 S. R. Finch, Analysis of a Radical Expansion, Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.
%H A105816 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 A105816 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 A105816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NestedRadicalConstant.html">Nested Radical Constant.</a>
%H A105816 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tirukkannapuram_Vijayaraghavan">Tirukkannapuram Vijayaraghavan</a>
%F A105816 continued fraction representation of: sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n)=A001358(n))))).
%e A105816 2.66352563480685654498944673272195514599922982689272932914833705868...
%t A105816 fQ[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]] == 2; t = Select[ Range[ 300], fQ[ # ] &]; f[n_] := Block[{k = n, s = 0}, While[k > 0, s = Sqrt[s + t[[k]]]; k-- ]; s]; ContinuedFraction[ f[90], 99] (* _Robert G. Wilson v_, Apr 21 2005 *)
%Y A105816 Cf. A001358; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105816, A239349 for other nested radicals.
%Y A105816 From _Robert G. Wilson v_: (Start)
%Y A105816 Cf. A072449, Decimal expansion of limit of a nested radical, sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ...
%Y A105816 Cf. A083869, a(1)=1 then a(n) is the least k>=1 such that the nested radical sqrt(a(1)^2+sqrt(a(2)^2+sqrt(a(3)^2+(....+sqrt(a(n)^2)))...) is an integer.
%Y A105816 Cf. A099874, Decimal expansion of a nested radical: cubeRoot(1 + cubeRoot(2 + cubeRoot(3 + cubeRoot(4 + ...
%Y A105816 Cf. A099876, Decimal expansion of a nested radical: sqrt(1! + sqrt(2! + sqrt(3! + ...
%Y A105816 Cf. A099877, Decimal expansion of a nested radical: sqrt(1^2 + cubeRoot(2^3 + 4thRoot(3^4 + 5thRoot(4^5 + ...
%Y A105816 Cf. A099878, Decimal expansion of a nested radical: sqrt(1 + cubeRoot(2 + 4thRoot(3 + 5thRoot(4 + ...
%Y A105816 Cf. A099879, Decimal expansion of a nested radical: sqrt(1^2 + sqrt(2^2 + sqrt(3^2 + ...
%Y A105816 (End)
%K A105816 cofr,nonn
%O A105816 0,1
%A A105816 _Jonathan Vos Post_, Apr 21 2005
%E A105816 Offset changed by _Andrew Howroyd_, Aug 03 2024