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A239349
Decimal expansion of prime version of Ramanujan's infinite nested radical.
19
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, 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, 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
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
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.
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
Sequence in context: A355283 A198111 A194562 * A198675 A203131 A259484
KEYWORD
cons,nonn
AUTHOR
Jonathan Sondow, Mar 16 2014
STATUS
approved