A030178 -id:A030178 - cp's OEIS frontend

A332915 Decimal expansion of the constant W(1) + 1/W(1), where W is Lambert's function.

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

Offset: 1

Martin Renner, Mar 02 2020

The graph of the exponential function exp(x) moved to the right by W(1) + 1/W(1) touches the graph of the natural logarithm log(x) at point (x,y) = (1/W(1), W(1)) = (A030797, A030178).

			2.33036612476168058322517043916206263018983377385398...

Cf. A030178, A030797, A052888.

evalf[200](LambertW(1) + 1/LambertW(1));

RealDigits[N[LambertW[1] + 1/LambertW[1], 120]][[1]] (* Vaclav Kotesovec, Mar 02 2020 *)

my(x=lambertw(1)); x+1/x \\ Michel Marcus, Mar 02 2020

Equals A030178 + A030797 = A030178 + 1/A030178 = 1/A030797 + A030797.

Equals 2 + Integral_{x=0..1} W(x) dx. - Amiram Eldar, Jul 18 2021

A352771 Decimal expansion of the unique real solution to exp(x) = 1/x - 1.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Apr 02 2022

			0.40105813754154703565062537500645662909560698650459...

István Mező, The Lambert W Function, Its Generalizations and Applications, CRC Press, 2022.

István Mező and Árpád Baricz, On the generalization of the Lambert W function with applications in theoretical physics, arXiv:1408.3999 [math.CA], 2014-2015.
István Mező and Árpád Baricz, On the generalization of the Lambert W function, Trans. Amer. Math. Soc., Vol. 369, No. 11 (2017), pp. 7917-7934.
Index entries for transcendental numbers

Cf. A030178, A048993.

RealDigits[x /. FindRoot[Exp[x] == 1/x - 1, {x, 1}, WorkingPrecision -> 120]][[1]]

solve(x=0.1, 1, exp(x) - 1/x + 1) \\ Michel Marcus, Apr 02 2022

Equals W_1(1), where W_1(x) is the 1-Lambert function.
Equals 1/2 + Sum_{n>=2} (Sum_{k=1..n-1} ((n+k-1)!/(n-1)!) * Stirling2(n-1,k)*(-1/2)^k)/(2^n*n!).
Both formulas are from Mező and Baricz (2017).

cp's OEIS Frontend

A332915 Decimal expansion of the constant W(1) + 1/W(1), where W is Lambert's function.

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