A101314 -id:A101314 - cp's OEIS frontend

A226278 Decimal expansion of the number x > 1 defined by 2*log(x) = x - 1.

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

Offset: 1

José María Grau Ribas, Jun 02 2013

There are two solutions to the equation 2*log(x) = x - 1: {1, 3.51286...}.

Apart from the leading digit the same as A201890. - R. J. Mathar, Jun 05 2013

			x = 3.512862417252339353965475233218432653832833663402647422251789454...

Cf. A101314, A201890, A202343.

Digits := 100; evalf([solve(2*ln(n)=n-1,n)]);

RealDigits[x /. FindRoot[2*Log[x] == x - 1, {x, 3.5}, WorkingPrecision -> 110]][[1]]
RealDigits[N[Exp[-ProductLog[-1,-1/(2*Sqrt[E])]-1/2],110]][[1]] (* Natalia L. Skirrow, Jul 13 2025 *)

solve(x=3,4,2*log(x)-x+1) \\ Charles R Greathouse IV, Jun 05 2013

Equals 1 + A201890.
Equals exp(-LambertW_-1(-1/(2*sqrt(e)))-1/2). - Natalia L. Skirrow, Jul 13 2025
Equals 1/A101314 = exp(A202343). - Hugo Pfoertner, Jul 13 2025

A226604 Decimal expansion of the maximum value reached by the function x-x^2+x^2 log(x) in the interval [0,1].

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Jun 13 2013

Equals (c-c^2)/2 where c = A101314.

			0.1018160943972684375348232152630558119587523815313957473674364483728932...

Cf. A101314.

EE = -1/2/ ProductLog[-1, -1/2/Sqrt[E]]; RealDigits[N[(EE - EE^2)/2, 100]][[1]]

Offset corrected by Rick L. Shepherd, Jan 01 2014

cp's OEIS Frontend

A226278 Decimal expansion of the number x > 1 defined by 2*log(x) = x - 1.

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