OFFSET
1,1
COMMENTS
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....
Converges by Vijayaraghavan's test or Herschfeld's test, together with the Prime Number Theorem. - Petros Hadjicostas and Jonathan Sondow, Mar 23 2014
REFERENCES
S. Ramanujan, J. Indian Math. Soc., III (1911), 90 and IV (1912), 226.
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
A. Herschfeld, On Infinite Radicals, Amer. Math. Monthly, 42 (1935), 419-429.
Jonathan Sondow and Petros 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.
Wikipedia, Tirukkannapuram Vijayaraghavan.
FORMULA
Equals 2*sqrt(1 + 3*sqrt(1 + 5*sqrt(1 + 7*sqrt(1 + 11*sqrt(1 + ...))))).
Lim_{n->infinity} 2*sqrt(1 + 3*sqrt(1 + 5*sqrt(1 + ... + prime(n)*sqrt(1))))).
sqrt(4 + sqrt(144 + sqrt(129600 + ...))) = sqrt(A(1) + sqrt(A(2) + sqrt(A(3) + ...))), where A = A239350 = superprimorials squared.
EXAMPLE
9.4050436124452175781376337429786005794187565225902363965922...
MATHEMATICA
RealDigits[ Fold[ #2*Sqrt[ 1 + #1] &, 0, Reverse[ Prime[ Range[ 400]]]], 10, 110][[1]]
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Jonathan Sondow, Mar 16 2014
STATUS
approved