A072449 -id:A072449 - cp's OEIS frontend

A105816 Continued fraction expansion of the semiprime nested radical (A105815).

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

Offset: 0

Jonathan Vos Post, Apr 21 2005

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].

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

			2.66352563480685654498944673272195514599922982689272932914833705868...

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.

Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
J. Sondow and P. 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.
Wikipedia, Tirukkannapuram Vijayaraghavan

Cf. A001358; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105816, A239349 for other nested radicals.
From Robert G. Wilson v: (Start)
Cf. A072449, Decimal expansion of limit of a nested radical, sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ...
Cf. 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.
Cf. A099874, Decimal expansion of a nested radical: cubeRoot(1 + cubeRoot(2 + cubeRoot(3 + cubeRoot(4 + ...
Cf. A099876, Decimal expansion of a nested radical: sqrt(1! + sqrt(2! + sqrt(3! + ...
Cf. A099877, Decimal expansion of a nested radical: sqrt(1^2 + cubeRoot(2^3 + 4thRoot(3^4 + 5thRoot(4^5 + ...
Cf. A099878, Decimal expansion of a nested radical: sqrt(1 + cubeRoot(2 + 4thRoot(3 + 5thRoot(4 + ...
Cf. A099879, Decimal expansion of a nested radical: sqrt(1^2 + sqrt(2^2 + sqrt(3^2 + ...
(End)

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] (* Robert G. Wilson v, Apr 21 2005 *)

continued fraction representation of: sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n)=A001358(n))))).

Offset changed by Andrew Howroyd, Aug 03 2024

A277313 Decimal expansion of the nested logarithm log(1+log(2+log(3+log(4+...)))).

Original entry on oeis.org

8, 2, 0, 3, 5, 9, 8, 6, 2, 2, 0, 8, 7, 8, 9, 7, 8, 8, 4, 7, 3, 4, 6, 6, 7, 9, 4, 9, 4, 0, 6, 3, 9, 1, 5, 8, 4, 1, 5, 9, 0, 9, 7, 5, 3, 4, 1, 3, 1, 6, 1, 9, 3, 7, 6, 5, 4, 6, 8, 7, 6, 7, 4, 9, 4, 8, 5, 0, 2, 4, 0, 7, 0, 1, 9, 2, 2, 9, 3, 8, 4, 6, 3, 2, 4, 5, 1, 7, 7, 4, 5, 4, 4, 7, 9, 2, 9, 9, 2, 8, 8, 2, 9, 8, 2

Offset: 0

Alex Klotz, Oct 09 2016

Found empirically. Logarithms are natural.
Converges to within 10^-4 of the asymptotic value when the innermost term is 7. The first fifteen digits after the decimal point can be found numerically by using 17 nested terms.
No closed form expression is known. Probably transcendental but this is unproved.
Empirically, the number of bits of precision with N as the innermost term is 0.02N^2 + 2.24N - 8.5. This means that using N as the largest innermost term gives (0.02N^2 + 2.24N - 8.5)*(log_10(2)) digits. - Cade Brown, Oct 10 2016

			0.82035986220878978847346679494...

Cade Brown, Table of n, a(n) for n = 0..14997

Similar in concept to A072449.

Cf. A278812 (log(2*log(3*log(4*...))), or log(2) + log(log(3) + log(log(4) + ...))).

// Computes b bits, and uses MPFR for multiprecision.
#include 
#include 
#include 
int main() {
    int b=256, i;
    int N = 500 + (int)(4 * floor(-56+sqrt(3561+50*b)));
    mpfr_t m;
    mpfr_init2(m, b);
    mpfr_set_ui(m, N, rnd);
    for (i = N; i > 0; --i) {
        mpfr_log(m, m, MPFR_RNDN);
        mpfr_add_ui(m, m, i - 1, MPFR_RNDN);
    }
    mpfr_printf("\nval %.*Rf\n\n", b - 10, m);
    mpfr_clear(m);
} /* Cade Brown, Oct 10 2016 */

x=100;
for i=99:-1:1
x=log(i+x);
end
%the initial value of x can be increased for greater precision, but it converges starting well below 100

RealDigits[SequenceLimit[N[Table[Log[Fold[#2 + Log[#1] &, Reverse@Range[n]]], {n, 1, 100}], 200]], 10, 105][[1]] (* Vladimir Reshetnikov, Oct 11 2016 *)
RealDigits[ Fold[ Log[#1 + #2] &, 0, Reverse[ Range[74]]], 10, 111][[1]] (* Robert G. Wilson v, Oct 26 2016 *)

More digits from Alois P. Heinz, Oct 09 2016

A083729 Decimal expansion of sqrt(2)/(sqrt(2)-1)^2 = 3*sqrt(2)+4.

Original entry on oeis.org

8, 2, 4, 2, 6, 4, 0, 6, 8, 7, 1, 1, 9, 2, 8, 5, 1, 4, 6, 4, 0, 5, 0, 6, 6, 1, 7, 2, 6, 2, 9, 0, 9, 4, 2, 3, 5, 7, 0, 9, 0, 1, 5, 6, 2, 6, 1, 3, 0, 8, 4, 4, 2, 1, 9, 5, 3, 0, 0, 3, 9, 2, 1, 3, 9, 7, 2, 1, 9, 7, 4, 3, 5, 3, 8, 6, 3, 2, 1, 1, 1, 6, 5, 5, 1, 1, 6, 2, 6, 0, 2, 9, 8, 2, 9, 2, 4, 7, 1, 8, 2, 0, 5, 0, 4

Offset: 1

Benoit Cloitre, Jun 15 2003

			8.242640687119285146405...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A010474, A072449.

SetDefaultRealField(RealField(100)); 4 + 3*Sqrt(2); // G. C. Greubel, Sep 11 2018

RealDigits[4 + 3*Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Sep 11 2018 *)

default(realprecision, 100); 4 + 3*sqrt(2) \\ G. C. Greubel, Sep 11 2018

Equals A010474 plus 4. - R. J. Mathar, Sep 17 2008

Equals Sum_{i>=1} i/sqrt(2)^i. - Christoph B. Kassir, Aug 30 2021

A296547 Decimal expansion of (2 + (3 + (4 + (5 + ...)^(1/5))^(1/4))^(1/3))^(1/2).

Original entry on oeis.org

1, 9, 1, 1, 6, 3, 9, 2, 1, 6, 2, 4, 5, 8, 2, 4, 2, 8, 3, 8, 8, 3, 3, 0, 9, 5, 3, 7, 6, 4, 0, 6, 2, 4, 6, 6, 4, 6, 0, 8, 5, 8, 1, 5, 3, 9, 4, 0, 6, 2, 2, 3, 8, 2, 4, 2, 9, 6, 6, 7, 9, 9, 2, 4, 4, 2, 5, 9, 6, 3, 2, 5, 6, 4, 5, 1, 7, 5, 7, 8, 5, 2, 8, 6, 9, 8, 4, 1, 7, 2, 7, 3, 8, 4, 1, 0, 5, 8, 5, 8

Offset: 1

Ilya Gutkovskiy, Dec 15 2017

			1.9116392162458242838833095376406246646085815394...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A072449, A296301.

A072450 Continued fraction expansion of the limit of a nested radical, sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... )))).

Original entry on oeis.org

1, 1, 3, 7, 1, 1, 1, 2, 3, 1, 4, 1, 1, 2, 1, 2, 20, 1, 2, 2, 2, 1, 2, 1, 1, 6, 2, 8, 5, 13, 2, 3, 1, 1, 115, 1, 4, 38, 4, 3, 1, 2, 1, 1, 1, 14, 1, 10, 4, 4, 5, 2, 2, 3, 19, 1, 1, 1, 5, 2, 1, 4, 1, 3, 1, 3, 4, 1, 8, 47, 33, 1, 1, 5, 13, 1, 14, 1, 5, 1, 1, 2, 17, 2, 1, 108, 9, 16, 3, 1, 2, 2, 3, 1, 5, 6, 2

Offset: 0

Robert G. Wilson v, Aug 01 2002

Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(4 + ... = 1.75793275661800...

Increasing partial continued fractions of the above are 1, 3, 7, 20, 115, 233, 301, 328, 16902, ...

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, London, England, 1997, page 30.

Eric Weisstein's World of Mathematics, Nested Radical.

Cf. A072449 (decimal expansion).

ContinuedFraction[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Range[100]]], 100]

Offset changed by Andrew Howroyd, Jul 06 2024

A105818 Continued fraction expansion of the Fibonacci nested radical (A105817).

Original entry on oeis.org

1, 1, 1, 1, 23, 18, 1, 1, 1, 1, 1, 1, 2, 1, 22, 2, 1, 53, 1, 1, 10, 1, 1, 17, 2, 4, 1, 27, 1, 2, 422, 3, 3, 13, 12, 5, 28, 1, 3, 1, 2, 1, 3, 2, 4, 6, 6, 3, 5, 50, 1, 1, 6, 3, 2, 1, 118, 2, 1, 1, 2, 6, 1, 4, 1, 1, 5, 2, 3, 3, 16, 1, 4, 6, 2, 2, 22, 4, 3, 10, 1, 1, 49, 5, 1, 1, 12, 1, 1, 3, 13, 3, 10, 1, 2

Offset: 0

Jonathan Vos Post, Apr 21 2005

The decimal expansion of this is A105817. "It was discovered by 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]. We know the asymptotic limit of Fibonacci numbers is Phi^n (Binet expansion) and that Phi^n < 2^n and hence that the Fibonacci Nested Radical converges.

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

			1.66198246232781155796760608181513129505616756246503500829906806743...

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
S. R. Finch, "Analysis of a Radical Expansion." Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.

Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
J. Sondow and P. 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.
Wikipedia, Tirukkannapuram Vijayaraghavan

Cf. A000045; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105815, A105816, A105817, A239349 for other nested radicals.

f[n_] := Block[{k = n, s = 0}, While[k > 0, s = Sqrt[s + Fibonacci[k]]; k-- ]; s]; ContinuedFraction[ f[46], 95] (* Robert G. Wilson v, Apr 21 2005 *)

Sqrt(1 + Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(5 + ... + Sqrt(Fibonacci(n) = A000045)))).

Offset changed by Andrew Howroyd, Aug 03 2024

A283749 Decimal expansion of the limit of the nested sin(1+ sin(2+ sin(3+ sin(4+ ...)))).

Original entry on oeis.org

9, 9, 4, 1, 6, 6, 6, 7, 8, 1, 2, 0, 6, 7, 6, 3, 3, 8, 6, 9, 0, 6, 2, 1, 7, 8, 8, 6, 9, 5, 5, 5, 8, 5, 1, 2, 8, 2, 5, 2, 2, 9, 7, 4, 3, 4, 3, 1, 9, 2, 2, 9, 8, 6, 5, 0, 4, 7, 0, 8, 7, 9, 0, 3, 2, 2, 4, 6, 5, 8, 8, 4, 8, 6, 7, 8, 8, 8, 4, 6, 8, 8, 4, 6, 6, 3, 0, 5, 5, 3, 7, 6, 3, 5, 1, 9, 5, 4, 0, 1, 5, 1, 6, 5, 1, 4, 7, 9, 8, 7, 0

Offset: 0

Alex Klotz and Robert G. Wilson v, Mar 15 2017

In radians.
No closed form expression is known.
Probably transcendental but this has not been proved.
By Lindemann's theorem, at most one of sin(1+ sin(2+ sin(3+...))) and sin(2+ sin(3+ sin(4+ ...))) is algebraic. - Robert Israel, Mar 15 2017

			0.9941666781206763386906217886955585128252297434319229865047087903224658848...

Cf. A072449, A277313.

RealDigits[Fold[ Sin[#1 + #2] &, 0, Reverse[Range[284]]], 10, 111][[1]]

A296041 Decimal expansion of sqrt(1 + sqrt(3 + sqrt(6 + sqrt(10 + sqrt(15 + ... + sqrt(k*(k + 1)/2 + ...)))))).

Original entry on oeis.org

1, 8, 6, 4, 4, 5, 8, 9, 5, 8, 1, 6, 3, 4, 8, 8, 1, 3, 2, 3, 5, 2, 0, 0, 3, 7, 1, 5, 2, 7, 3, 9, 4, 3, 7, 8, 4, 1, 5, 6, 4, 2, 2, 0, 6, 9, 8, 2, 6, 6, 4, 9, 9, 2, 3, 5, 8, 6, 9, 0, 4, 3, 8, 1, 8, 2, 7, 5, 3, 2, 5, 7, 3, 5, 8, 3, 4, 5, 5, 8, 1, 7, 7, 9, 9, 6, 4, 4, 9, 0, 6, 1, 9, 8, 3, 7, 8, 2, 2, 2

Offset: 1

Ilya Gutkovskiy, Dec 03 2017

			1.86445895816348813235200371527394378415642206982664...

Eric Weisstein's World of Mathematics, Nested Radical

Cf. A000217, A072449, A099876, A099879, A105546, A105817, A239349, A257574, A257575, A257576, A257577, A257578.

A296140 Decimal expansion of 1/sqrt(1 + 1/sqrt(2 + 1/sqrt(3 + 1/sqrt(4 + 1/sqrt(5 + ...))))).

Original entry on oeis.org

7, 8, 3, 7, 6, 6, 3, 0, 9, 2, 3, 6, 3, 9, 6, 4, 6, 9, 9, 5, 1, 9, 4, 3, 0, 7, 7, 6, 3, 8, 7, 4, 2, 8, 1, 2, 7, 0, 4, 1, 4, 1, 1, 8, 0, 7, 7, 3, 8, 7, 7, 4, 7, 5, 5, 8, 9, 6, 6, 0, 7, 6, 0, 5, 7, 5, 2, 6, 6, 5, 3, 9, 7, 7, 4, 0, 1, 4, 0, 9, 4, 7, 5, 4, 4, 3, 5, 1, 6, 4, 2, 9, 4, 7, 1, 9, 9, 6, 1, 2

Offset: 0

Ilya Gutkovskiy, Dec 05 2017

			0.7837663092363964699519430776387428127041411807738774755...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Nested Radical Constant

Cf. A052119, A060006, A072449.

RealDigits[Fold[1/Sqrt[#1 + #2]&, 0, Range[100, 1, -1]], 10, 100][[1]] (* Jean-François Alcover, Dec 19 2017 *)

A301514 Infinite nested radical negated: isqrt(1+2isqrt(1+3isqrt(1+4isqrt(1+5i*sqrt(1+...))))).

Original entry on oeis.org

2, 4, 9, 4, 6, 2, 5, 1, 4, 7, 5, 4, 1, 0, 0, 9, 2, 9, 0, 1, 8, 7, 2, 2, 7, 1, 3, 0, 0, 0, 4, 0, 7, 7, 4, 5, 1, 2, 6, 8, 7, 7, 3, 3, 6, 6, 7, 0, 8, 4, 1, 3, 9, 6, 1, 4, 7, 5, 5, 9, 2, 3, 0, 0, 4, 4, 3, 4, 3, 3, 9, 6, 4, 9, 9, 1, 2, 2, 0, 9, 9, 2, 5, 3, 2, 8, 2, 9, 3, 2, 1, 4, 9, 6, 5, 0, 9, 7, 8, 9, 3, 0, 1, 2, 6

Offset: 1

Robert G. Wilson v, Dec 15 2018

Inspired by Ramanujan's infinite nested radical of natural numbers.

			2.4946251475410092901872271300040774512687733667084139614755923004434339649...

Ramanujan's 2nd Notebook, Chapter XII, Page 108, entry 4.

Robert G. Wilson v, Table of n, a(n) for n = 1..10001

Cf. A072449, A239349.

RealDigits[Chop[N[-I*Fold[#2*Sqrt[1 + #1*I] &, 0, Reverse[Range[400]]], 125]], 10, 120][[1]] (* Vaclav Kotesovec, Dec 22 2018 *)

b(n) = imag(fold((x,y)->sqrt(1+x*y), Vecrev([2..n+1])*I));
N=106; default(realprecision, N); limitnum(b,,4) \\ Gheorghe Coserea, Dec 23 2018

cp's OEIS Frontend

A105816 Continued fraction expansion of the semiprime nested radical (A105815).

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A277313 Decimal expansion of the nested logarithm log(1+log(2+log(3+log(4+...)))).

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A083729 Decimal expansion of sqrt(2)/(sqrt(2)-1)^2 = 3*sqrt(2)+4.

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A296547 Decimal expansion of (2 + (3 + (4 + (5 + ...)^(1/5))^(1/4))^(1/3))^(1/2).

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A072450 Continued fraction expansion of the limit of a nested radical, sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... )))).

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Mathematica

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A105818 Continued fraction expansion of the Fibonacci nested radical (A105817).

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A283749 Decimal expansion of the limit of the nested sin(1+ sin(2+ sin(3+ sin(4+ ...)))).

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A296041 Decimal expansion of sqrt(1 + sqrt(3 + sqrt(6 + sqrt(10 + sqrt(15 + ... + sqrt(k*(k + 1)/2 + ...)))))).

A301514 Infinite nested radical negated: isqrt(1+2isqrt(1+3isqrt(1+4isqrt(1+5i*sqrt(1+...))))).