A073229 -id:A073229 - cp's OEIS frontend

A194346 Decimal expansion of h_o(1/17), where h_o(x) is the odd infinite power tower function.

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

Offset: 0

Jonathan Sondow, Aug 27 2011

The odd infinite power tower function is h_o(x) = lim f(n,x) as n --> infinity, where f(n+1,x) = x^x^(f(n,x)) and f(1,x) = x. The even infinite power tower function h_e(x) is the same limit except with f(1,x) = x^x (see A194347). The limits exist if and only if 0 < x <= e^(1/e). If (1/e)^e <= x <= e^(1/e), then h_o(x) = h_e(x) = h(x) (the infinite power tower function-see the comments in A073230) and y = h(x) is a solution of x^y = y. If 0 < x < (1/e)^e, then h_o(x) < h_e(x), and two solutions of x^x^y = y are y = h_o(x) and y = h_e(x). For example, y = h_o(1/16) = 1/4 and y = h_e(1/16) = 1/2 are solutions of (1/16)^(1/16)^y = y.

h_o(1/17) and h_e(1/17) are irrational, and at least one of them is transcendental (see Sondow and Marques 2010).

			0.204274736665518499175698745186446957991668690348422572736592466759324966133336...

See the References in Sondow and Marques 2010.

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see Definition 4.3, Figure 7, and top of p. 163.

Cf. A073229, A073230, A073243, A194347.

a = N[1/17, 100]; Do[a = (1/17)^(1/17)^a, {3000}]; RealDigits[a, 10, 100] // First
RealDigits[ Fold[ N[#2^#1, 128] &, 1/17, Table[1/17, {5710}]], 10, 105][[1]] (* Robert G. Wilson v, Mar 20 2012 *)

solve(x=0,1,17^(-17^-x)-x) \\ Charles R Greathouse IV, Mar 20 2012

A037087 Beatty sequence for e^(1/e).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 93, 95

Offset: 0

Felice Russo

nonn

Cf. A073229.

Floor[Range[0, 100]*Exp[1/E]] (* Paolo Xausa, Jul 05 2024 *)

a(n) = floor(n*e^(1/e)) = floor(n*A073229).

A093157 Decimal expansion of inflection point of x^(1/x).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Mar 25 2004

The smaller of the two real roots of 1 - 3*x + log(x)*(log(x) + 2*x - 2) = 0. - Amiram Eldar, Jun 17 2021

			0.581932705...

Eric Weisstein's World of Mathematics, Steiner's Problem.

Cf. A073229, A103476.

f[x_] := x^(1/x); RealDigits[ x /. FindRoot[ f''[x] == 0, {x, 1/2}, WorkingPrecision -> 102]] // First (* Jean-François Alcover, Feb 08 2013 *)

A103476 Decimal expansion of second inflection point of x^(1/x).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Feb 07 2005

The larger of the two real roots of 1 - 3*x + log(x)*(log(x) + 2*x - 2) = 0. - Amiram Eldar, Jun 17 2021

			4.36777096...

Eric Weisstein's World of Mathematics, Steiner's Problem.

Cf. A073229, A093157.

RealDigits[ x /. FindRoot[D[x^(1/x), {x, 2}] == 0, {x, 4}, WorkingPrecision -> 110], 10, 102] // First (* Jean-François Alcover, Feb 07 2013 *)

A133334 Signature sequence of e^(1/e) - the solution (the y value) to Steiner's problem: find the max value attained by y=x^(1/x).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 8, 1, 4, 7, 3, 6, 9, 2, 5, 8, 1, 4, 7, 10, 3, 6, 9, 2, 5, 8, 11, 1, 4, 7, 10, 3, 6, 9, 12, 2, 5, 8, 11, 1, 4, 7, 10, 13, 3, 6, 9, 12, 2, 5, 8, 11, 14, 1, 4, 7, 10, 13, 3, 6, 9, 12, 15, 2, 5, 8, 11, 14, 1, 4, 7, 10, 13, 16

Offset: 1

Gregg Whisler, Oct 19 2007

nonn

T. D. Noe, Table of n, a(n) for n=1..1000
C. Kimberling, Fractal sequences
Index entries for sequences related to signature sequences

Cf. A073229.

Take[Transpose[ Sort[Flatten[Table[{i + j*(E^(1/E)), i}, {i, 17}, {j, 15}], 1], #1[[1]] < #2[[1]] &]][[2]], 91]

A175993 Decimal expansion of e^(1+1/e), e = exp(1).

Original entry on oeis.org

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

Offset: 1

Dylan Hamilton, Nov 05 2010

Area of the rectangle having as opposite corners the origin and (e,e^(1/e)), maximum of the graph y=x^(1/x).

			3.92701439474164490992795352226729686971604001234684620191649848504155615457299800255983...

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001113, A073229.

First[RealDigits[Exp[1 + 1/E], 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

default(realprecision,99);e=exp(1);e^(1+1/e)  \\ M. F. Hasler, Nov 27 2012

Product of A001113 and A073229.

Edited by M. F. Hasler, Nov 27 2012

A350358 Value of -F(0), where F(x) is the indefinite integral of x^(1/x).

Original entry on oeis.org

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

Offset: 0

Robert B Fowler, Dec 26 2021

The indefinite integral of x^(1/x) can be derived by expanding
    x^(1/x) = exp(log(x)/x) = Sum_{n>=0} (log(x)/x)^n/n!,
then integrating term-by-term to get the double summation
    F(x) = x + (1/2)*(log(x))^2
             - (Sum_{n>=2} (n-1)^(-n-1) * x^(-n+1)/n!
             * (Sum_{k=0..n} A350297(n,k)*(log(x))^k)).
We assume the constant of integration in F(x) is zero.
To compute definite integrals of x^(1/x) for ranges of x starting at x=0 or x=1, we would need the values
    F(0) = lim {x->0} F(x) = -0.4203695887832022981324...
    F(1) = 1 - Sum_{n>=2} (n-1)^(-n-1) = -0.06687278808178024266...
Note that the definite Integral_{t=0..1} x^(1/x) = F(1)-F(0) = A175999.
The calculation of F(0) requires some care. See the FORMULA below.
Since x^(1/x) is the inverse of the infinite exponentiation function E(y) = y^(y^(y^(...))) = x, the definite integrals of these two functions are related by
    (Integral_{t=0..y} E(t) dt) + (Integral_{t=0..x} t^(1/t) dt) = x*y.
Note that the repeated exponentiation in E(y) converges for 0 <= y < e^(1/e), but diverges for y > e^(1/e).

			0.4203695887832022981324...

Peter Luschny, Table of n, a(n) for n = 0..999
Jonathan Sondow and Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see Figure 5.
Eric Weisstein's World of Mathematics, Steiner's Problem.

Cf. A350297, A175999, A176000, A073229.

# The chosen parameters give about 100 exact decimal places.
using Nemo
RR = RealField(1100)
function F(b::Int, x::arb)
    lnx = log(x)
    s = sum(gamma_regularized(RR(n+2), RR(n)*lnx) * RR(n)^(-n-2) for n in 1:b)
    x + (lnx * lnx) / RR(2) - s
end
println( F(1400, RR(0.015)) ) # Peter Luschny, Dec 26 2021

# The chosen parameters give about 100 exact decimal places.
Digits := 400: F := proc(b, x) local s, lnx; lnx := log(x);
   s := add(add((n*lnx)^k / k!, k = 0..n+1) / (n*x)^(n+2), n = 1..b);
   x - x^2 * s + lnx^2 / 2 end:
F(2000, 0.01); # Peter Luschny, Dec 27 2021

RealDigits[N[Sum[1/n^(n+2), {n, 1, 100}] + Integrate[x^(1/x), {x, 0, 1}] - 1, 110]][[1]] (* Amiram Eldar, Dec 29 2021 *)

The calculation of F(0) requires some care, because terms in the formula for F(x) diverge for x=0, but converge for all x > 0, although convergence is progressively slower as x approaches zero. To calculate F(0), first choose a desired precision d (absolute error). Then choose any x such that 0 < x^(1/x) < d, and evaluate F(x) as defined above.
Since F(x)-F(0) < x^(1/x) < d, F(0)=F(x) to the desired precision.
For small x, the main summation terms initially increase in absolute value (but with alternating signs), reach a maximum of about 1/d at n = -log(x)/x, then decrease at an accelerating rate and reach a value of around d at n = -3.5911*log(x)/x, at which point the large terms have mostly cancelled out, and further terms are below precision level.
For example, for a final precision of d = 10^-50, the calculation must allow for intermediate terms and sums as large as 1/d = 10^50, so these terms must be evaluated to at least 100 digits. For 50 digits, F(.03) is a suitable choice, because .03^(1/.03) = 1.7273...*10^-51 < 10^-50. A few extra digits should also be allowed for round off error.

More terms from Hugo Pfoertner, Dec 26 2021

A175994 Decimal expansion of the definite integral of y=x^(1/x) for x=0 to e, the only maximum of this graph.

Original entry on oeis.org

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

Offset: 1

Dylan Hamilton, Nov 05 2010

			2.6618257053804178284970393376513958302149708209833035482146784850914702106571...

J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see Figure 5.

Cf. A073229 (decimal expansion of e^(1/e)).

RealDigits[ NIntegrate[ x^(1/x), {x, 0, E}, WorkingPrecision -> 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

A175996 Decimal expansion of (e-1)*(e^(1/e)-1).

Original entry on oeis.org

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

Offset: 0

Dylan Hamilton, Nov 05 2010

Area of rectangle having as opposite corners (1,1) and the unique maximum of the graph y = x^(1/x), namely (e, e^(1/e)).

			0.764064705...

Cf. A073229

RealDigits[(E-1)(E^(1/E)-1),10,120][[1]] (* Harvey P. Dale, Feb 16 2014 *)

A175997 Decimal expansion of the definite integral of y=x^(1/x)-1 for x=1 to e, the only maximum of this graph.

Original entry on oeis.org

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

Offset: 0

Dylan Hamilton, Nov 05 2010

			0.5900470762199505376709154960920350873674669264745699794399029327595933957066...

Cf. A073229 (decimal expansion of e^(1/e)).

RealDigits[NIntegrate[x^(1/x)-1,{x,1,E},WorkingPrecision->250]][[1]] (* Harvey P. Dale, Mar 05 2016 *)

Corrected by Harvey P. Dale, Mar 05 2016

cp's OEIS Frontend

A194346 Decimal expansion of h_o(1/17), where h_o(x) is the odd infinite power tower function.

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A037087 Beatty sequence for e^(1/e).

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A093157 Decimal expansion of inflection point of x^(1/x).

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A103476 Decimal expansion of second inflection point of x^(1/x).

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A133334 Signature sequence of e^(1/e) - the solution (the y value) to Steiner's problem: find the max value attained by y=x^(1/x).

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A175993 Decimal expansion of e^(1+1/e), e = exp(1).

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A350358 Value of -F(0), where F(x) is the indefinite integral of x^(1/x).

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A175994 Decimal expansion of the definite integral of y=x^(1/x) for x=0 to e, the only maximum of this graph.

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