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%I #31 May 14 2020 11:17:42
%S 9,4,0,5,0,4,3,6,1,2,4,4,5,2,1,7,5,7,8,1,3,7,6,3,3,7,4,2,9,7,8,6,0,0,
%T 5,7,9,4,1,8,7,5,6,5,2,2,5,9,0,2,3,6,3,9,6,5,9,2,2,1,7,2,1,8,5,6,0,6,
%U 8,5,9,4,2,4,2,2,1,9,9,1,2,9,8,7,3,7,7,4,0,1,4,1,0,4,9,2,9,0,6,2,8,5,5,8,9,1,8,2,6,9
%N Decimal expansion of prime version of Ramanujan's infinite nested radical.
%C Replace each factor n = 1, 2, 3, ... with prime(n) = 2, 3, 5, ... in Ramanujan's infinite nested radical 1*sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + ...))) = 3, obtaining 2*sqrt(1 + 3*sqrt(1 + 5*sqrt(1 + ...))) = 9.405043....
%C Converges by Vijayaraghavan's test or Herschfeld's test, together with the Prime Number Theorem. - _Petros Hadjicostas_ and _Jonathan Sondow_, Mar 23 2014
%D S. Ramanujan, J. Indian Math. Soc., III (1911), 90 and IV (1912), 226.
%D T. Vijayaraghavan, in Collected Papers of Srinivasa Ramanujan, G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, eds., Cambridge Univ. Press, 1927, p. 348; reprinted by Chelsea, 1962.
%H G. C. Greubel, <a href="/A239349/b239349.txt">Table of n, a(n) for n = 1..1000</a>
%H A. Herschfeld, <a href="http://www.jstor.org/stable/2301294">On Infinite Radicals</a>, Amer. Math. Monthly, 42 (1935), 419-429.
%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 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tirukkannapuram_Vijayaraghavan">Tirukkannapuram Vijayaraghavan</a>.
%F Equals 2*sqrt(1 + 3*sqrt(1 + 5*sqrt(1 + 7*sqrt(1 + 11*sqrt(1 + ...))))).
%F Lim_{n->infinity} 2*sqrt(1 + 3*sqrt(1 + 5*sqrt(1 + ... + prime(n)*sqrt(1))))).
%F sqrt(4 + sqrt(144 + sqrt(129600 + ...))) = sqrt(A(1) + sqrt(A(2) + sqrt(A(3) + ...))), where A = A239350 = superprimorials squared.
%e 9.4050436124452175781376337429786005794187565225902363965922...
%t RealDigits[ Fold[ #2*Sqrt[ 1 + #1] &, 0, Reverse[ Prime[ Range[ 400]]]], 10, 110][[1]]
%Y Cf. A105546, A239350.
%K cons,nonn
%O 1,1
%A _Jonathan Sondow_, Mar 16 2014
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