A105815 -id:A105815 - 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

A105546 Decimal expansion of prime nested radical.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Apr 12 2005

sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ...)))) = 1.75793275661800...
"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 log(a_n)/2^n exists." [Clawson, 229; cf. A072449].
We know the asymptotic limit of primes and hence that the Prime 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

			2.10359749633989726261993964968532544404216228824001387298687284563...

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.

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. A000040, A072449, A239349.
A105548 is the continued fraction representation of this prime nested radical.
A105815 is the similar semiprime nested radical.
A105817 is the Fibonacci nested radical.

RealDigits[Fold[Sqrt[#1 + #2] &, 0, Reverse[Prime[Range[ 80]]]], 10, 111][[1]] (* Robert G. Wilson v, May 31 2005 *)

sqrt(2 + sqrt(3 + sqrt(5 + sqrt(7 + sqrt(11 + ... + sqrt(prime(n) + ...)))).

Crossrefs corrected by Jaroslav Krizek, Jan 03 2015

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.

			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)))).

A118835 Numerators of n-th convergent to continued fraction with semiprime terms.

Original entry on oeis.org

4, 25, 229, 2315, 32639, 491900, 10362539, 228467758, 5722056489, 149001936472, 4922785960065, 167523724578682, 5868253146213935, 223161143280708212, 8709152841093834203, 400844191833597081550, 19650074552687350830153, 1002554646378888489419353, 55160155625391554268894568

Offset: 1

Jonathan Vos Post, May 01 2006

Denominators are A118836. A118835/A118836 converges to semiprime continued fraction constant ~ 4.1636688. The first fractions are 4, 25/6, 229/55, 2315/556, 32639/7839, 491900/118141, 10362539/2488800, 228467758/54871741, 5722056489/1374282325, 149001936472/35786212191, 4922785960065/1182319284628, 167523724578682/40234641889543, 5868253146213935/1409394785418633.

These are to semiprimes as A001040 are to natural numbers. See also A105815 Decimal expansion of the semiprime nested radical.

			a(1) = 4 = numerator of 4/1.
a(2) = 25 = numerator of 25/6 = 4+1/6.
a(3) = 229 = numerator of 229/55 = 4+1/(6+1/9).
a(4) = 2315 = numerator of 2315/556 = 4+1/(6+1/(9+(1/10))).

Eric Weisstein's World of Mathematics, Continued Fraction Constant.

Cf. A001358, A001040, A001053, A105815, A118836.

sp = Select[Range[10^3], PrimeOmega[#] == 2 &]; Numerator@ Table[ FromContinuedFraction[ Take[sp, i]], {i, 20}] (* Giovanni Resta, Jun 16 2016 *)

a(n) = numerator of continued fraction [4; 6, 9, 10, 14, ... A001358(n)]. CONTINUANT transform of A001358.

Data corrected by Giovanni Resta, Jun 16 2016

A118836 Denominators of n-th convergent to continued fraction with semiprime terms.

Original entry on oeis.org

1, 6, 55, 556, 7839, 118141, 2488800, 54871741, 1374282325, 35786212191, 1182319284628, 40234641889543, 1409394785418633, 53597236487797597, 2091701617809524916, 96271871655725943733, 4719413412748380767833, 240786355921823145103216, 13247968989113021361444713

Offset: 1

Jonathan Vos Post, May 01 2006

Numerators are A118835. A118835/A118836 converges to semiprime continued fraction constant ~ 4.1636688. The first fractions are 4, 25/6, 229/55, 2315/556, 32639/7839, 491900/118141, 10362539/2488800, 228467758/54871741, 5722056489/1374282325, 149001936472/35786212191, 4922785960065/1182319284628, 167523724578682/40234641889543, 5868253146213935/1409394785418633.

These are to semiprimes as A001053 are to natural numbers. See also A105815 Decimal expansion of the semiprime nested radical.

			a(1) = 1 = denominator of 4/1.
a(2) = 6 = denominator of 25/6 = 4+1/6.
a(3) = 55 = denominator of 229/55 = 4+1/(6+1/9).
a(4) = 556 = denominator of 2315/556 = 4+1/(6+1/(9+(1/10))).

Eric Weisstein's World of Mathematics, Continued Fraction Constant.

Cf. A001358, A001040, A001053, A105815, A118835.

sp = Select[Range[10^3], PrimeOmega[#] == 2 &]; Denominator @ Table[ FromContinuedFraction[ Take[sp, i]], {i, 20}] (* Giovanni Resta, Jun 16 2016 *)

a(n) = denominator of continued fraction [4; 6, 9, 10, 14, ... A001358(n)]. CONTINUANT transform of A001358.

Corrected and edited by Giovanni Resta, Jun 16 2016

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

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

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A105546 Decimal expansion of prime nested radical.

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A105817 Decimal expansion of the Fibonacci nested radical.

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A118835 Numerators of n-th convergent to continued fraction with semiprime terms.

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A118836 Denominators of n-th convergent to continued fraction with semiprime terms.

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

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