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

Author

Jonathan Sondow, Mar 16 2014

Keywords

• cons
• nonn

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

Examples

			9.4050436124452175781376337429786005794187565225902363965922...
		

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.

Crossrefs

Cf. A105546, A239350.

Programs

Mathematica

RealDigits[ Fold[ #2*Sqrt[ 1 + #1] &, 0, Reverse[ Prime[ Range[ 400]]]], 10, 110][[1]]

Formula

Equals 2*sqrt(1 + 3*sqrt(1 + 5*sqrt(1 + 7*sqrt(1 + 11*sqrt(1 + ...))))).

Equals lim_{n->oo} 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.