A073084 -id:A073084 - cp's OEIS frontend

A344390 Continued fraction for A073084, the constant LambertW(log(sqrt(2)))/log(sqrt(2)).

0, 1, 3, 3, 1, 1, 563, 4, 24, 1, 1, 1, 1, 8, 1, 1, 2, 1, 1, 1, 3, 2, 3, 2, 1, 1, 2, 190, 1, 1, 2, 1, 1, 2, 1, 6, 11, 3, 1, 1, 1, 1, 2, 1, 2, 2, 1, 4, 1, 1, 65, 1, 1, 1, 11, 25, 1, 2, 2, 2, 3, 29, 2, 16, 2, 3, 17, 5, 3, 4, 1, 3, 3, 20, 3, 1, 1, 2, 1, 2, 2, 2, 1, 3, 105, 8, 17, 1, 5, 1

Offset: 0

Jianing Song, May 17 2021

a(6) = 563 shows that x = -23/30 is a good approximation to the negative solution to 2^x = x^2.

			0.76666469596212309311... = 0 + 1/(1 + 1/(3 + 1/(3 + 1/(1 + 1/(1 + 1/(563 + ...))))))

Cf. A073084.

ContinuedFraction[ProductLog[Log[Sqrt[2]]]/Log[Sqrt[2]], 100]

default(realprecision, 100); contfrac(lambertw(log(sqrt(2)))/log(sqrt(2)))

A010503 Decimal expansion of 1/sqrt(2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

The decimal expansion of sqrt(50) = 5*sqrt(2) = 7.0710678118654752440... gives essentially the same sequence.
Also real and imaginary part of the square root of the imaginary unit. - Alonso del Arte, Jan 07 2011
1/sqrt(2) = (1/2)^(1/2) = (1/4)^(1/4) (see the comments in A072364).
If a triangle has sides whose lengths form a harmonic progression in the ratio 1 : 1/(1 + d) : 1/(1 + 2d) then the triangle inequality condition requires that d be in the range -1 + 1/sqrt(2) < d < 1/sqrt(2). - Frank M Jackson, Oct 11 2011
Let s_2(n) be the sum of the base-2 digits of n and epsilon(n) = (-1)^s_2(n), the Thue-Morse sequence A010060, then Product_{n >= 0} ((2*n + 1)/(2*n + 2))^epsilon(n) = 1/sqrt(2). - Jonathan Vos Post, Jun 03 2012
The square root of 1/2 and thus it follows from the Pythagorean theorem that it is the sine of 45 degrees (and the cosine of 45 degrees). - Alonso del Arte, Sep 24 2012
Circumscribed sphere radius for a regular octahedron with unit edges. In electrical engineering, ratio of effective amplitude to peak amplitude of an alternating current/voltage. - Stanislav Sykora, Feb 10 2014
Radius of midsphere (tangent to edges) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014
Positive zero of the Hermite polynomial of degree 2. - A.H.M. Smeets, Jun 02 2025

			0.7071067811865475...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Sections 1.1, 7.5.2, and 8.2, pp. 1-3, 468, 484, 487.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Harry J. Smith, Table of n, a(n) for n = 0..20000
P. C. Fishburn and J. A. Reeds, Bell inequalities, Grothendieck's constant and root two, SIAM J. Discrete Math., Vol. 7, No. 1, Feb. 1994, pp. 48-56.
Ovidiu Furdui, Problem 1, Problem Corner, Research Group in Mathematical Inequalities and Applications, 2010.
Michael Penn, A surprisingly convergent limit, YouTube video, 2022.
Michael Penn, The infinite fraction of your dreams (nightmare?), YouTube video, 2022.
Jonathan Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; arXiv:1108.6096 [math.NT], 2011, see p. 3 in the link.
Eric Weisstein's World of Mathematics, Digit Product.
Wikipedia, Platonic solid.
Donald R. Woods, Problem E 2692, Elementary Problems, The American Mathematical Monthly, Vol. 85, No. 1 (1978), p. 48; A Transcendental Function Satisfy a Duplication Formula, by David Robbins, ibid., Vol. 86, No. 5 (1979), pp. 394-395.
Index entries for algebraic numbers, degree 2.

Cf. A000120, A040042, A072364, A268682.
Cf. A073084 (infinite tetration limit).
Platonic solids circumradii: A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).
Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A019863 (icosahedron), A239798 (dodecahedron).
Cf. A000217, A010060, A019812, A019824, A019851, A019857, A019884, A019896.

1/Sqrt(2); // Vincenzo Librandi, Feb 21 2016

Digits:=100; evalf(1/sqrt(2)); Wesley Ivan Hurt, Mar 27 2014

N[ 1/Sqrt[2], 200]
RealDigits[1/Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 25 2019 *)

default(realprecision, 20080); x=10*(1/sqrt(2)); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b010503.txt", n, " ", d)); \\ Harry J. Smith, Jun 02 2009

1/sqrt(2) = cos(Pi/4) = sqrt(2)/2. - Eric Desbiaux, Nov 05 2008
a(n) = 9 - A268682(n). As constants, this sequence is 1 - A268682. - Philippe Deléham, Feb 21 2016
From Amiram Eldar, Jun 29 2020: (Start)
Equals sin(Pi/4) = cos(Pi/4).
Equals Integral_{x=0..Pi/4} cos(x) dx. (End)
Equals (1/2)*A019884 + A019824 * A010527 = A019851 * A019896 + A019812 * A019857. - R. J. Mathar, Jan 27 2021
Equals hypergeom([-1/2, -3/4], [5/4], -1). - Peter Bala, Mar 02 2022
Limit_{n->oo} (sqrt(T(n+1)) - sqrt(T(n))) = 1/sqrt(2), where T(n) = n(n+1)/2 = A000217(n) is the triangular numbers. - Jules Beauchamp, Sep 18 2022
Equals Product_{k>=0} ((2*k+1)/(2*k+2))^((-1)^A000120(k)) (Woods, 1978). - Amiram Eldar, Feb 04 2024
From Stefano Spezia, Oct 15 2024: (Start)
Equals 1 + Sum_{k>=1} (-1)^k*binomial(2*k,k)/2^(2*k) [Newton].
Equal Product_{k>=1} 1 - 1/(4*(2*k - 1)^2). (End)
Equals Product_{k>=0} (1 - (-1)^k/(6*k+3)). - Amiram Eldar, Nov 22 2024

More terms from Harry J. Smith, Jun 02 2009

A104748 Decimal expansion of solution to x*2^x = 1.

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Mar 23 2005

Writing the equation as (1/2)^x = x, the solution is the value of the infinite power tower function h(t) = t^t^t^... at t = 1/2. The solution is a transcendental number. - Jonathan Sondow, Aug 29 2011

Equals LambertW(log(2))/log(2) since, for 1/E^E <= c < 1, c^c^c^... = LambertW(log(1/c))/log(1/c). - Stanislav Sykora, Nov 03 2013

			x = 0.641185744504985984486200482114823666562820957191101... = (1/2)^(1/2)^(1/2)^...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see p. 160.
Wikipedia, Lambert W function
Index entries for transcendental numbers

Equals 1/A030798.

Cf. A073084.

RealDigits[ ProductLog[ Log[2]]/Log[2], 10, 111][[1]] (* Robert G. Wilson v, Mar 23 2005 *)
RealDigits[x/.FindRoot[x 2^x==1,{x,.6},WorkingPrecision->100]][[1]] (* Harvey P. Dale, Apr 17 2019 *)

lambertw(log(2))/log(2) \\ Stanislav Sykora, Nov 03 2013

More terms from Robert G. Wilson v, Mar 23 2005

Offset corrected by R. J. Mathar, Feb 05 2009

A166928 Decimal expansion of smaller solution to 3^x = x^3.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Oct 23 2009

The larger solution is of course 3.
Also, the limit of infinite tetration a^a^...^a of a=3^(1/3) (=A002581), i.e., lim_{n->oo} x(n) where x(n+1)=a^x(n), x(1)=a. - M. F. Hasler, Nov 03 2013
The constant is transcendental (Vassilev-Missana, p. 23). - Peter Bala, Jan 01 2014

			2.47805268028830241189373651689...

M. Vassilev-Missana, Some results on infinite power towers, Notes on Number Theory and Discrete Mathematics, Vol. 16 (2010) No. 3, 18-24.
Index entries for transcendental numbers

Cf. A073084, A341328.

RealDigits[ -3*ProductLog[ -Log[3]/3 ] / Log[3], 10, 105] // First (* Jean-François Alcover, Mar 05 2013 *)

PARI
```
solve(x=2,exp(1),3^x-x^3)
```

A341328 Decimal expansion of the smaller solution (i.e., the solution other than x = 5) to 5^x = x^5.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Feb 09 2021

Also decimal expansion of the other solution to log(x)/x = log(5)/5.
Also the limit of infinite tetration a^a^...^a, where a = 5^(1/5).
Let b be a rational number > e, then: if b is not of the form b = (1 + 1/s)^(s+1) for some positive integer s, then the other solution to b^x = x^b (or equivalently, log(x)/x = log(b)/b) is transcendental. In particular, if b is a positive integer other than 1, 2 and 4, then the other solution to b^x = x^b is transcendental (Vassilev-Missana, p. 23).

			If x = 1.7649219145257758827587235909114591014..., then log(x)/x = log(5)/5.

M. Vassilev-Missana, Some results on infinite power towers, Notes on Number Theory and Discrete Mathematics, Vol. 16 (2010) No. 3, 18-24.
Index entries for transcendental numbers

Cf. A073084, A166928.

RealDigits[-5*ProductLog[-Log[5]/5]/Log[5], 10, 105]
RealDigits[x/.FindRoot[5^x==x^5,{x,1.7},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Jan 22 2023 *)

default(realprecision, 92); solve(x=1, 2, 5^x-x^5)

Equals -(5/log(5))*W(-log(5)/5), where W is the principal branch of the Lambert W function.

A344905 Decimal expansion of the solution to x^x = sqrt(2).

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Jun 01 2021

			1.304351178901036533647201231486234...

Cf. A030798, A073084.

RealDigits[Log[Sqrt[2]]/ProductLog[Log[Sqrt[2]]], 10, 100][[1]] (* Amiram Eldar, Jun 02 2021 *)
RealDigits[x/.FindRoot[x^x==Sqrt[2],{x,1},WorkingPrecision-> 120],10,120][[1]] (* Harvey P. Dale, Jun 18 2021 *)

solve(x=1,2,x^x-sqrt(2)) \\ Hugo Pfoertner, Jun 02 2021

Equals log(2)/(2*LambertW(log(2)/2)). - Alois P. Heinz, Jun 02 2021

Equals 1/A073084. - Jason Bard, Aug 20 2025

A266092 Decimal expansion of the power tower of 1/sqrt(3): the real solution to 3^(x/2)*x = 1.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

			(1/sqrt(3))^(1/sqrt(3))^(1/sqrt(3))^(1/sqrt(3))^… = 0.686026724536251319713006846182…

Eric Weisstein's World of Mathematics, Power Tower
Eric Weisstein's World of Mathematics, Lambert W-Function

Cf. A020760, A030178, A073084, A073243, A231096.

RealDigits[(2 ProductLog[Log[3]/2])/Log[3], 10, 120][[1]]

t=log(3)/2; lambertw(t)/t \\ Charles R Greathouse IV, Apr 18 2016

Equals 2*LambertW(log(3)/2)/log(3).

A362738 a(n) is the least nonnegative integer solution for y such that x > 1 is an integer in the equation n^y*x^n = n^(x^(1/n)).

Original entry on oeis.org

0, 0, 192, 3000, 46440, 823200, 16776704, 387419760, 9999999000, 285311669280, 8916100446528, 302875106590056, 11112006825555272, 437893890380856000, 18446744073709547520, 827240261886336759264, 39346408075296537569592, 1978419655660313589117120, 104857599999999999999992000

Offset: 2

Thomas Scheuerle, May 01 2023

Corresponding solutions for x are 256 (2^(2^3)) at n = 2, 19683 (3^(3^2)) at n = 3 and n^(n^2) for all n > 3.

Further solution pairs are of the form y = n^(n^k) - n^(2+k) and x = n^(n^(1+k)) with k > 0.

Cf. A000312, A058126, A073084.

PARI
```
a(n) = max(0,n^n-n^3)
```

E.g.f.: 1/(1 + LambertW(-x)) - x*(1 + 3*x + x^2)*exp(x) + 2*x^2, where LambertW() is the Lambert W-function.

a(n) = n^n - n^3 for n > 2.

A363229 Decimal expansion of e^(-2*LambertW(-log(2)/4)).

Original entry on oeis.org

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

Offset: 1

Thomas Scheuerle, May 22 2023

The least real solution of x^2 = 2^sqrt(x). This equation has two real solutions the other is 256.
Let x be this constant, and c = 2*log(x)/log(2); then c^4 = 2^c.
Let x be this constant, and c = 1/sqrt(x); then c^c = 1/2^(1/4).

			1.5366769...

Cf. A073084, A104750.

RealDigits[E^(-2 ProductLog[-Log[2]/4]),10,100][[1]]

PARI
```
\p 200
exp(-2*lambertw(-log(2)/4))
```

import math; from sympy import LambertW
print([i for i in str("%.30f" % math.exp(-2*LambertW(-math.log(2)/4)))])
# Javier Rivera Romeu, May 22 2023

Equals e^(-2*Sum_{k>=1} ((-k)^(-1+k)*(-log(2)/4)^k/k!)).
Equals e^(t*log(2)/2) where t = (2^(1/4))^(2^(1/4))^(2^(1/4))^(2^(1/4))^... is the infinite power tower over 2^(1/4).
Equals 16*LambertW(-log(2)/4)^2 / log(2)^2. - Vaclav Kotesovec, May 22 2023

cp's OEIS Frontend

A344390 Continued fraction for A073084, the constant LambertW(log(sqrt(2)))/log(sqrt(2)).

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A010503 Decimal expansion of 1/sqrt(2).

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A104748 Decimal expansion of solution to x*2^x = 1.

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A166928 Decimal expansion of smaller solution to 3^x = x^3.

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A341328 Decimal expansion of the smaller solution (i.e., the solution other than x = 5) to 5^x = x^5.

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A344905 Decimal expansion of the solution to x^x = sqrt(2).

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A266092 Decimal expansion of the power tower of 1/sqrt(3): the real solution to 3^(x/2)*x = 1.

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