A099878 -id:A099878 - cp's OEIS frontend

A099879 Decimal expansion of a nested radical: sqrt(1^2 + sqrt(2^2 + sqrt(3^2 + ...

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

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 03 2004

			1.94265542276398732822141329141266723768807363...

Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

Cf. A072449, A099874, A099876 to A099878 for other nested radicals.

k = 64; r = 65; While[k > 0, r = Sqrt[k^2 + r]; k-- ]; RealDigits[r, 10, 111][[1]] (* Robert G. Wilson v, Nov 04 2004 *)

t=0; forstep(n=100,1,-1,t=sqrt(t+n^2)); print(t)

\\ We need about b/log(b) steps, where epsilon = 2^-b.
my(b=bitprecision(1.),t); forstep(n=b\log(b)+9,1,-1, t=sqrt(t+n^2)); t \\ Charles R Greathouse IV, Aug 19 2025

A105817 Decimal expansion of the Fibonacci nested radical.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Apr 21 2005

The continued fraction expression of this is A105818. "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.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.
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, A105818, A239349 for other nested radicals.

Cf. A151558.

RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Fibonacci[ Range[50]]]], 10, 111][[1]] (* Robert G. Wilson v, Apr 21 2005 *)

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

A105815 Decimal expansion of the semiprime nested radical.

Original entry on oeis.org

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

Offset: 1

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]. The continued fraction representation of this constant is A105816.

			2.66352563480685654498944673272195514599922982689272932914833705868...

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.

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.
Wikipedia, Tirukkannapuram Vijayaraghavan.

For other nested radicals, see A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105816, A239349.

Cf. A001358.

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 *)

Limit_{n -> infinity} sqrt(4 + sqrt(6 + sqrt(9 + sqrt(10 + sqrt(14 + ... + sqrt(semiprime(n))))), where semiprime(n) = A001358(n).

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

Original entry on oeis.org

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

			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

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.

			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

cp's OEIS Frontend

A099879 Decimal expansion of a nested radical: sqrt(1^2 + sqrt(2^2 + sqrt(3^2 + ...

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A105817 Decimal expansion of the Fibonacci nested radical.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A105815 Decimal expansion of the semiprime nested radical.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions