A239349 Decimal expansion of prime version of Ramanujan's infinite nested radical.

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

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

Cf. A105546, A239350.

Sequence in context: A355283 A198111 A194562 * A198675 A203131 A259484

Adjacent sequences: A239346 A239347 A239348 * A239350 A239351 A239352

Keyword

cons,nonn

Author

Jonathan Sondow, Mar 16 2014

Status

approved