A073229 -id:A073229 - cp's OEIS frontend

A176000 Decimal expansion of 1 - A175999.

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

Offset: 0

Dylan Hamilton, Nov 05 2010

Area contained by y=x^(1/x),x=1, and y=0

Cf. A073229, A175999.

A194347 Decimal expansion of h_e(1/17), where h_e(x) is the even infinite power tower function.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Aug 27 2011

See the Comments, References and Links in A194346.

			0.560596485316498211176108570378470772301099831523122840744442447202319227572291...

Cf. A073229, A073230, A073243, A194346.

a = N[(1/17)^(1/17), 100]; Do[a = (1/17)^(1/17)^a, {3000}]; RealDigits[a, 10, 100] // First

A234604 Floor of the solutions to c = exp(1 + n/c) for n >= 0, using recursion.

Original entry on oeis.org

2, 3, 4, 4, 5, 6, 6, 7, 17, 35, 62, 103, 164, 256, 391, 589, 880, 1303, 1919, 2814, 4112, 5993, 8716, 12655, 18353, 26591, 38499, 55710, 80583, 116523, 168453, 243485, 351889, 508506, 734776, 1061672, 1533938, 2216216

Offset: 0

Richard R. Forberg, Dec 28 2013

nonn

For n = 1 to 7 recursion produces convergence to single valued solutions.
For n >= 8 a dual-valued oscillating recursion persists between two stable values. The floor of the upper value for each n is included here. (The lower values of c are under 6 and approach exp(1) = 2.71828 for large n.)
At large n, the ratio of a(n)/a(n-1) approaches exp(1/exp(1)) = 1.444667861009 with more digits given by A073229.
At n = 0, c = exp(1).
At n = 1, c = 3.5911214766686 = A141251.
At n = 2, c = 4.3191365662914
At n = 3, c = 4.9706257595442
At n = 4, c = 5.5723925978776
At n = 5, c = 6.1383336446072
At n = 6, c = 6.6767832796664
At n = 7, c = 7.1932188286406
The convergence becomes "dual-valued" at n > exp(2) = 7.3890560989 = A072334.
At values of n = 7 and 8 the convergence is noticeably slower than at either larger or smaller values of n.
The recursion at n = exp(2) is only "quasi-stable" where c reluctantly approaches exp(2) = exp(1 + exp(2)/exp(2)) from any starting value, but never reaches it, and is not quite able to hold it if given the solution, due to machine rounding errors.

Cf. A073229, A141251.

a(n) = floor(c) for the solutions to c = exp(1 + n/c) at n = 0 to 7, and the floor of the stable upper values of c for n >= 8.

Conjecture: a(n) = floor(e^(-e^(t^2/e^t - t)*t^2 + t + 1)) for all n > 13. - Jon E. Schoenfield, Jan 11 2014

Corrected and edited by Jon E. Schoenfield, Jan 11 2014

A277067 Decimal expansion of value of x such that the solution y to the equation x^y = y has equal real and imaginary parts.

Original entry on oeis.org

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

Offset: 0

David D. Acker, Sep 27 2016

It is not known if this number has a closed form.

			-0.750045256460151711237852993036822415525210961075147250937205...

Cf. A042972, A073229, A277069.

FindRoot[Re[-ProductLog[-Log[x]]/Log[x]] - Im[-ProductLog[-Log[x]]/Log[x]], {x, -0.76, -0.74}, WorkingPrecision -> 261]

The solution to x^y=y is y=-ProductLog(-log(x))/log(x).

A335028 Decimal expansion of Pi*(exp(1/e) - 1)/2.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, May 20 2020

The value of an integral (see formula) first calculated by Cauchy in 1825 (with an error that was corrected in 1826).

			0.69848264271788427226723035849771244456284836693292...

Harold P. Boas, Cauchy’s Residue Sore Thumb, The American Mathematical Monthly, Vol. 125, No. 1 (2018), pp. 16-28, preprint, arXiv:1701.04887v1 [math.HO], 2017.
Augustin-Louis Cauchy, Mémoire sur les intégrales définies prises entre des limites imaginaires, Paris, 1825, p. 65, equation 24.
Augustin-Louis Cauchy, Exercices de mathématiques, Paris, 1826, p. 108, equation 48.

Cf. A000796 (Pi), A001113 (e), A019609 (Pi*e), A019610(Pi*e/2), A073229 (e^(1/e)), A335027.

RealDigits[Pi*(Exp[1/E] - 1)/2, 10, 100][[1]]

Pi*(exp(1/exp(1)) - 1)/2 \\ Michel Marcus, May 20 2020

Equals Integral_{x=0..oo} (exp(cos(x)) * sin(sin(x)) * x /(x^2 + 1)) * dx.

A356562 Decimal expansion of the unique positive real root of the equation x^x^x = x^x + 1.

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Aug 12 2022

This constant is case m=2 in a family of real roots of x^^(m+1) - x^^m = 1, where ^^ is tetration. These roots have lim_{m->inf} x(m) = e^(1/e) (see A073229).

			1.67129219798893255...

Cf. A073229, A124930 (m=1).

RealDigits[x /. FindRoot[x^(x^x) == x^x + 1, {x, 2}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Aug 13 2022 *)

solve(x=1, 2, x^x^x - x^x - 1) \\ Michel Marcus, Aug 13 2022

A072551 Decimal expansion of sqrt(e^(1/e)) = 1.20194336847031...

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Aug 05 2002

This constant is related to the convergence properties of the following simple algorithm: w(n+2) = A^( w(n+1) + w(n) ) where A is a positive real. Take any w(1), w(2) reals>0, then w(n) converges if and only if, 0 < A < sqrt(e^(1/e)). For example if A=1/2 w(n) converges to 1/2, if A=1/3, w(n) converges to 0.408004405...(If w(n) converges the limit L is always independent of initial values w(1),w(2) and L is < e).

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 448-452.

A084574 Continued fraction expansion of e^(1/e).

Original entry on oeis.org

1, 2, 4, 55, 27, 1, 1, 16, 9, 3, 2, 8, 3, 2, 1, 1, 4, 1, 9, 6, 4, 1, 2, 1, 1, 1, 3, 2, 2, 770, 1, 2, 2, 4, 1, 4, 1, 2, 1, 4, 1, 1, 14, 107, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 1, 34, 12, 1, 4, 23, 4, 2, 1, 4, 1, 1, 1, 1, 1, 5, 3, 2, 346, 1, 2, 2, 1, 4, 1, 2, 2, 1, 1, 2, 2, 4, 1, 4, 3, 11, 1, 4, 60, 2, 56

Offset: 0

Paul D. Hanna, May 31 2003

Cf. A073229 (decimal expansion).

ContinuedFraction[E^(1/E),100] (* Harvey P. Dale, Aug 09 2016 *)

Offset changed by Andrew Howroyd, Jul 08 2024

A175995 Decimal expansion of A175993 minus A175994.

Original entry on oeis.org

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

Offset: 1

Dylan Hamilton, Nov 05 2010

Area of the region bounded by y=x^(1/x), x=0 and y=e^(1/e). [Corrected by Osea Fracchia, Jul 09 2023]

			1.2651886893612... = 3.927014394741644... - 2.6618257053804...

Cf. A073229, A175993, A175994.

A175998 Decimal expansion of (e-1)*(e^(1/e)-1) - int(x^(1/x)-1, x=1..e).

Original entry on oeis.org

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

Offset: 0

Dylan Hamilton, Nov 05 2010

Area contained by y=x^(1/x), x=e, and y=1.

Difference between the numbers defined in A175996 and A175997.

			0.174017629052883003238411446226169...

Cf. A073229, A175996, A175997.

e=exp(1); (e-1)*(e^(1/e)-1)-intnum(x=1,e,x^(1/x)-1)  \\ - M. F. Hasler, Nov 27 2012

cp's OEIS Frontend

A176000 Decimal expansion of 1 - A175999.

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A194347 Decimal expansion of h_e(1/17), where h_e(x) is the even infinite power tower function.

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A234604 Floor of the solutions to c = exp(1 + n/c) for n >= 0, using recursion.

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A277067 Decimal expansion of value of x such that the solution y to the equation x^y = y has equal real and imaginary parts.

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A335028 Decimal expansion of Pi*(exp(1/e) - 1)/2.

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A356562 Decimal expansion of the unique positive real root of the equation x^x^x = x^x + 1.

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A072551 Decimal expansion of sqrt(e^(1/e)) = 1.20194336847031...

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A084574 Continued fraction expansion of e^(1/e).

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A175995 Decimal expansion of A175993 minus A175994.

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