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Search: id:A105816
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| A105816 |
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Continued fraction expansion of the semiprime nested radical (A105815). |
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4
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| 2, 1, 1, 1, 34, 1, 2, 2, 2, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 9, 7, 1, 9, 1, 5, 1, 5, 1, 2, 7, 2, 2, 3, 5, 2, 1, 10, 8, 2, 3, 1, 1, 1, 12, 1, 1, 5, 4, 4, 2, 1, 1, 2, 2, 4, 13, 2, 2, 12, 3, 11, 15, 2, 2, 2, 23, 8, 1, 1, 3, 1, 2, 8, 19, 1, 5, 2, 7, 4, 1, 82, 22, 1, 1, 1, 2, 1, 1, 9, 1, 1, 1, 15, 8, 12, 2, 11, 1, 15
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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]. Robert G. Wilson v comments (for cross-references purposes): A072449 Decimal expansion of limit of a nested radical, Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(4 + ... A083869 a(1)=1 then a(n) is the least k>=1 such that the nested radical sqrt(a(1)^2+sqrt(a(2)^2+sqrt(a(3)^2+(....+sqrt(a(n)^2)))...) is an integer. A099874 Decimal expansion of a nested radical: CubeRoot(1 + CubeRoot(2 + CubeRoot(3 + CubeRoot(4 + ... A099876 Decimal expansion of a nested radical: sqrt(1! + sqrt(2! + sqrt(3! + ... A099877 Decimal expansion of a nested radical: Sqrt(1^2 + CubeRoot(2^3 + 4thRoot(3^4 + 5thRoot(4^5 + ... A099878 Decimal expansion of a nested radical: Sqrt(1 + CubeRoot(2 + 4thRoot(3 + 5thRoot(4 + ... A099879 Decimal expansion of a nested radical: sqrt(1^2 + sqrt(2^2 + sqrt(3^2 + ...
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REFERENCES
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Borwein, J. M. and de Barra, G. "Nested Radicals." Amer. Math. Monthly 98, 735-739, 1991.
Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 and 229.
S. R. Finch, Analysis of a Radical Expansion, Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.
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LINKS
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Eric Weisstein's World of Mathematics, Nested Radical Constant.
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FORMULA
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continued fraction representation of: Sqrt(4 + Sqrt(6 + Sqrt(9 + Sqrt(10 + Sqrt(14 + ... + Sqrt(Semiprime(n)=A001358(n))))).
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EXAMPLE
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2.66352563480685654498944673272195514599922982689272932914833705868...
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MATHEMATICA
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fQ[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]] == 2; t = Select[ Range[ 300], fQ[ # ] &]; f[n_] := Block[{k = n, s = 0}, While[k > 0, s = Sqrt[s + t[[k]]]; k-- ]; s]; ContinuedFraction[ f[90], 99] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 21 2005)
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CROSSREFS
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Cf. A001358; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105816 for other nested radicals.
Sequence in context: A067095 A070888 A067101 this_sequence A062979 A114781 A083890
Adjacent sequences: A105813 A105814 A105815 this_sequence A105817 A105818 A105819
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KEYWORD
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cofr,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 21 2005
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