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A105815 Decimal expansion of the semiprime nested radical. 6
2, 6, 6, 3, 5, 2, 5, 6, 3, 4, 8, 0, 6, 8, 5, 6, 5, 4, 4, 9, 8, 9, 4, 4, 6, 7, 3, 2, 7, 2, 1, 9, 5, 5, 1, 4, 5, 9, 9, 9, 2, 2, 9, 8, 2, 6, 8, 9, 2, 7, 2, 9, 3, 2, 9, 1, 4, 8, 3, 3, 7, 0, 5, 8, 6, 8, 0, 2, 3, 8, 8, 4, 8, 7, 9, 0, 3, 9, 3, 2, 9, 9, 3, 5, 6, 4, 3, 9, 6, 0, 5, 6, 8, 6, 4, 2, 4, 5, 5, 9, 9, 1, 4, 5, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The semiprime nested radical is defined by the infinite recursion: sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n))))). This converges by the criterion of T. Vijayaraghavan that "the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; Sloane A072449]. The continued fraction representation of this constant is A105816.
Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + ...))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, ... for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - Jonathan Sondow, Mar 25 2014
REFERENCES
Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
Steven R. Finch, Analysis of a Radical Expansion, Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, 2003, p. 8.
LINKS
Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
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.
Eric Weisstein's World of Mathematics, Nested Radical Constant.
FORMULA
Limit_{n -> infinity} sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n))))), where semiprime(n) = A001358(n).
EXAMPLE
2.66352563480685654498944673272195514599922982689272932914833705868...
MATHEMATICA
fQ[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]] == 2; RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Select[ Range[260], fQ[ # ] &]]], 10, 111][[1]] (* Robert G. Wilson v, May 31 2005 *)
CROSSREFS
For other nested radicals, see A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105816, A239349.
Cf. A001358.
Sequence in context: A011386 A097412 A153845 * A136696 A086358 A004152
KEYWORD
cons,nonn
AUTHOR
Jonathan Vos Post, Apr 21 2005
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)