A006890 -id:A006890 - cp's OEIS frontend

A257809 Lexicographically largest strictly increasing sequence of primes for which the continued square root map produces Feigenbaum's constant delta = 4.6692016... (A006890).

13, 67, 97, 139, 293, 661, 1163, 1657, 2039, 3203, 3469, 5171, 6361, 6661, 7393, 7901, 8969, 9103, 9137, 11971, 12301, 13487, 14083, 14699, 15473, 19141, 21247, 28099, 31039, 35423, 39047, 49223, 58427, 61493, 62171, 67699, 71971, 75869, 78857, 81533, 88007, 93199

Offset: 1

Chai Wah Wu, May 10 2015

nonn

The continued square root map takes a finite or infinite sequence (x, y, z, ...) to the number CSR(x, y, z,...) = sqrt(x + sqrt(y + sqrt(z + ...))). It is well defined if the logarithm of the terms is O(2^n).

The terms are defined to be the largest possible choice such that the sequence can remain strictly increasing without the CSR growing beyond delta = 4.66920...

			From _M. F. Hasler_, May 03 2018: (Start)
We look for a strictly increasing sequence of primes (p,q,r,...) such that CSR(p,q,r,...) = sqrt(p + sqrt(q + sqrt(r + ...))) = delta = 4.66920...
The first term must be less than delta^2 ~ 21.8, but p = 19 and also p = 17 are excluded, since CSR(17,19,23,...) > 4.67. It appears that p = 13 does not lead to a contradiction, so this is the largest possible choice for p, whence a(1) = 13.
The second term could be chosen to be q = 17, provided that subsequent terms are large enough to ensure CSR(p, q, r,...) = delta, which is always possible. But one can verify that any q between 19 and 67 is also possible without contradiction. If we try q = 71, then we find that CSR(13, 71, 73, ...) > 4.68. So a(2) = 67, etc. (End)

Chai Wah Wu, Table of n, a(n) for n = 1..1000
Popular Computing (Calabasas, CA), The CSR Function, Vol. 4 (No. 34, Jan 1976), pages PC34-10 to PC34-11. Annotated and scanned copy.
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.
Wikipedia, Feigenbaum constants.
1019 decimal digits of Feigenbaum's delta (due to David Broadhurst).

Cf. A006890, A257582, A257764, A257574.

(CSR(v,s)=forstep(i=#v,1,-1,s=sqrt(v[i]+s));s); a=[13]; for(n=1,50, print1(a[#a]","); for(i=primepi(a[#a])+1,oo, CSR(concat(a,vector(9,j,prime(i+j))))>=delta&& (a=concat(a,prime(i)))&& break)) \\  For delta, see A006890. - M. F. Hasler, May 03 2018

Edited by M. F. Hasler, May 02 2018

A159766 Continued fraction expansion of the Feigenbaum constant A006890 = 4.66920160910...

Original entry on oeis.org

4, 1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, 1, 33, 1, 1, 53, 1, 1, 1, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 239, 1, 3, 31, 1, 1, 11, 1, 13, 123, 2, 2, 2, 2, 13, 15, 1, 2, 3, 3, 1, 3, 1, 1, 6, 1, 3, 1, 1, 13, 8, 1, 7, 1, 2, 1, 8, 7, 1, 17, 1, 6, 1, 1, 3, 1, 1, 13, 1, 1, 4, 2, 9, 124, 1, 1, 3

Offset: 0

Harry J. Smith, Apr 21 2009

Feigenbaum bifurcation velocity delta. See A006890 for further information.

Apart from the first term, the same as A069544. - R. J. Mathar, Apr 28 2009

			4.66920160910299067185320382... = 4 + 1/(1 + 1/(2 + 1/(43 + 1/(2 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..964 [shortened by _Kevin Ryde_, Sep 30 2024]

Cf. A006890 (decimal expansion of the constant), A069544 (continued fraction of the fractional part), A069261 (Egyptian fraction).

fbvd = (* copy value from the link in A006890 or immediately above *); ContinuedFraction@ fbvd (* Robert G. Wilson v, Apr 27 2009 *)

{ default(realprecision,1019); delta=4.\
6692016091029906718532038204662016172581855774757686327456513430\
0413433021131473713868974402394801381716598485518981513440862714\
2027932522312442988890890859944935463236713411532481714219947455\
6443658237932020095610583305754586176522220703854106467494942849\
8145339172620056875566595233987560382563722564800409510712838906\
1184470277585428541980111344017500242858538249833571552205223608\
7250291678860362674527213399057131606875345083433934446103706309\
4520191158769724322735898389037949462572512890979489867683346116\
2688911656312347446057517953912204556247280709520219819909455858\
1946136877445617396074115614074243754435499204869180982648652368\
4387027996490173977934251347238087371362116018601281861020563818\
1835409759847796417390032893617143215987824078977661439139576403\
7760537119096932066998361984288981837003229412030210655743295550\
3888458497370347275321219257069584140746618419819610061296401614\
8771294441590140546794180019813325337859249336588307045999993837\
5411726563553016862529032210862320550634510679399023341675; A159766=contfrac(delta); for (n=1, 967, write("b159766.txt", n-1, " ", A159766[n])); } \\ Harry J. Smith, May 15 2009

A159766=contfrac(delta) \\ M. F. Hasler, Apr 30 2018

Old PARI program deleted by Harry J. Smith, May 19 2009

A102817 Decimal expansion of Gamma(delta)^2 where delta is the Feigenbaum bifurcation velocity constant (A006890).

Original entry on oeis.org

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

Offset: 3

Gerald McGarvey, Feb 26 2005

Let x be this constant, then Integral_{t=1..x} sin(t)/sqrt(t) dt = 0.655555692248871113068...
delta^2 = 21.8014436664499573..., (delta/Gamma(delta))^2 = 0.10000663312663433933000349...
If s is solution of Gamma(s) - sqrt(218) = 0 then 1/((s - delta)*Gamma(delta)^6) = 2.5555951358396... whereas a^(Pi/4) = 2.055596478435... where a is Feigenbaum alpha constant (A006891), the difference = 0.4999986574... ~ 1/(2 + 10^-5.27)
10*cos(Gamma(delta)^2) + Pi = -0.199999019922688714710053...

			217.99997644999881468628813957793609890726797890973...

D. Broadhurst, Feigenbaum constants to 1018 decimal places

Cf. A006890, A006891.

Set delta then RealDigits[Gamma[delta]^2, 10, 110][[1]]

acos(Pi/10+.0199999019922688714710053)+69*Pi \\ Yields ~ 30 digits. Using (2e5-1)/(1e7-1) yields ~ 15 digits. For a better value use, e.g., delta from the Broadhurst link. - M. F. Hasler, Apr 30 2018

A104123 Decimal expansion of the constant c = sqrt((137 - 1/(57+sqrt(Pi)/10))/(2*Pi)), an approximation to the Feigenbaum bifurcation velocity constant delta (A006890).

Original entry on oeis.org

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

Offset: 1

Gerald McGarvey, Mar 06 2005

c - delta = 0.000000000080501779597..., Gamma(delta) - InverseGamma(1/(c-delta)) = 0.50050037514..., log(log(1/(c-d))) - Pi = 0.00440025013324...

			4.669201609183492451450661894052330619516966105558694366297827...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A006890.

RealDigits[Sqrt[(137 - 1/(57 + Sqrt[Pi]/10))/(2 Pi)], 10, 110][[1]]

sqrt((137 - 1/(57 + sqrt(Pi)/10))/(2*Pi)) \\ G. C. Greubel, Jan 13 2017

A101419 Decimal expansion of d - log(d-e) where d is Feigenbaum's bifurcation velocity constant (A006890, delta constant) and e is Euler's constant.

Original entry on oeis.org

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

Offset: 1

Gerald McGarvey, Jan 16 2005

From Plouffe's Inverter: log(s)^2 * a^2 * sin(1)^1 = 4.000900694855076... where s is Sierpinski's constant (A062089) and a is Feigenbaum's reduction parameter (A006891, alpha constant). 20 + Pi - exp(Pi) = .00090002081...

			4.00090066535299100480962602422024957759530322082790880094053997689...

Cf. A006890, A006891, A001113, A062089.

A102819 Decimal expansion of (Gamma(delta)/delta)^2 where delta is the Feigenbaum bifurcation velocity constant (A006890).

Original entry on oeis.org

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

Offset: 1

Gerald McGarvey, Feb 26 2005

Let c be this constant, then c - 30*gamma^2 = .0039990171004565...

			9.999336731332016740224955368827864952665582544...

D. Broadhurst, Feigenbaum constants to 1018 decimal places

Cf. A006890.

Set delta then RealDigits[(Gamma[delta]/delta)^2, 10, 110][[1]]

A069261 Denominators of the Egyptian fraction for the fractional part of Feigenbaum's constant, 4.6692...

Original entry on oeis.org

2, 6, 395, 303319, 131209492876, 45596605913248081159007, 34243827483200809826686815883136413405197711755, 111445370519459209554489628949586784217535791333333948765270067675689059510906528783799426730444

Offset: 1

Christopher Lund (clund(AT)san.rr.com), Apr 14 2002

The next term in the series, a(9), is ~ 10^190.

The sequence gives the denominators for the fractional part of delta only. One could prefix four 1's in order to get (sum of reciprocals) = delta.

Kevin Ryde, Table of n, a(n) for n = 1..10

Cf. A006890 (Feigenbaum's constant), A069544 (continued fraction).

t=delta-4/*from A006890, or use: t=contfracpnqn(A069544); t[1,1]/t[2,1]*/; for(i=1,8,print1(1\t+1",");t-=1/(1\t+1)) \\ Requires delta to 93 decimals or A069544 to 90 terms (up to [...,1,1,4]) to get a(7) correctly, 180 terms for a(8). - M. F. Hasler, Apr 30 2018

a(n) = ceiling(1/(delta - 4 - Sum_{0 < i < n} 1/a(i))) is the smallest integer such that 4 + Sum_{i=1..n} 1/a(i) < delta = 4.6620... - M. F. Hasler, Apr 30 2018

Edited by M. F. Hasler, Apr 30 2018

A006891 Decimal expansion of Feigenbaum reduction parameter.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Colin Mallows, Jeffrey Shallit

			2.502907875095892822283902873218215786381271376727149977336192056779235...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 24.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 65-76
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 1..1018
Keith Briggs, A precise calculation of the Feigenbaum constants, Math. Comp., 57 (1991), 435-439.
Stuart Brorson, A High Precision Calculation of Feigenbaum's Alpha Using Julia, JuliaHub, 2017.
B. Derrida, A. Gervois and Y. Pomeau, Universal metric properties of bifurcations, J. Phys. A 12 (1979), 269-296.
Richard J. Mathar, Chebyshev series representation of Feigenbaum's period-doubling function, arXiv:1008.4608 [math.DS], 2010.
Simon Plouffe, Feigenbaum constants
Simon Plouffe, Plouffe's Inverter, Feigenbaum constants to 1018 decimal places
Judi Anne Thurlby, Rigorous calculations of renormalisation fixed points and attractors, Ph. D. Thesis, Univ. Portsmouth, (England, 2021). 400 digits in Section 3.8.
Eric Weisstein's World of Mathematics, Feigenbaum Constant
Wikipedia, Feigenbaum constants.

Cf. A006890 (Feigenbaum bifurcation velocity), A159767 (continued fraction).

More terms from Simon Plouffe, Jan 06 2002

A069544 Continued fraction for Feigenbaum's constant - 4 = 0.6692...

Original entry on oeis.org

1, 2, 43, 2, 163, 2, 3, 1, 1, 2, 5, 1, 2, 3, 80, 2, 5, 2, 1, 1, 1, 33, 1, 1, 53, 1, 1, 1, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 239, 1, 3, 31, 1, 1, 11, 1, 13, 123, 2, 2, 2, 2, 13, 15, 1, 2, 3, 3, 1, 3, 1, 1, 6, 1, 3, 1, 1, 13, 8, 1, 7, 1, 2, 1, 8, 7, 1, 17, 1, 6, 1, 1, 3, 1, 1, 13, 1, 1, 4, 2, 9, 124

Offset: 0

Christopher Lund (clund(AT)san.rr.com), Apr 17 2002

This sequence should have offset 1, since the first term already corresponds to the fractional and not to the integer part. See A159766 for a better variant. - M. F. Hasler, Apr 30 2018

Robert G. Wilson v, Table of n, a(n) for n = 0..952
David Broadhurst, 1019 decimal digits of Feigenbaum's delta. Correspondence to Simon Plouffe and others, 22-Mar-1999.

Cf. A159766 (continued fraction of delta), A006890 (decimal expansion).

default(realprecision,999);{/*paste here delta=... from the Broadhurst link*/};contfrac(delta)[^1] \\ M. F. Hasler, Apr 30 2018

delta - 4 = 1/( 1 + 1/( 2 + 1/( 43 + 1/( 2 + 1/( 163 + ... ))))) = 0.66920160910... - M. F. Hasler, Apr 30 2018

A108952 Decimal expansion of 1/delta, where delta = Feigenbaum constant.

Original entry on oeis.org

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

Offset: 0

Lekraj Beedassy, Jul 21 2005

In a dynamical system exhibiting period-doubling cascades,1/delta is the limiting ratio by which the nonlinear parameter's range of stable values shrinks after each successive onset of period-doubling.

			1/delta=0.21416937706232649247893481889316178341380

Cf. A006890.

Removed leading zero and adjusted offset - R. J. Mathar, Feb 06 2009

cp's OEIS Frontend

A257809 Lexicographically largest strictly increasing sequence of primes for which the continued square root map produces Feigenbaum's constant delta = 4.6692016... (A006890).

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A159766 Continued fraction expansion of the Feigenbaum constant A006890 = 4.66920160910...

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A102817 Decimal expansion of Gamma(delta)^2 where delta is the Feigenbaum bifurcation velocity constant (A006890).

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A104123 Decimal expansion of the constant c = sqrt((137 - 1/(57+sqrt(Pi)/10))/(2*Pi)), an approximation to the Feigenbaum bifurcation velocity constant delta (A006890).

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A101419 Decimal expansion of d - log(d-e) where d is Feigenbaum's bifurcation velocity constant (A006890, delta constant) and e is Euler's constant.

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A102819 Decimal expansion of (Gamma(delta)/delta)^2 where delta is the Feigenbaum bifurcation velocity constant (A006890).

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A069261 Denominators of the Egyptian fraction for the fractional part of Feigenbaum's constant, 4.6692...

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A006891 Decimal expansion of Feigenbaum reduction parameter.

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