A239354 -id:A239354

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A244009

Decimal expansion of 1 - log(2).

+10
15

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Fraction of numbers which are sqrt-smooth, see A048098 and A063539. - Charles R Greathouse IV, Jul 14 2014

Asymptotic survival probability in the 100 prisoners problem. - Alois P. Heinz, Jul 08 2022

REFERENCES

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

LINKS

Table of n, a(n) for n=0..105.

Peter Bala, A continued fraction expansion for the constant 1 - log(2)

Donald E. Knuth and Luis Trabb Pardo, Analysis of a simple factorization algorithm, Theoretical Computer Science 3:3 (1976), pp. 321-348.

The Ramanujan Machine, Using algorithms to discover new mathematics.

Wikipedia, 100 prisoners problem

Index entries for transcendental numbers

FORMULA

Equals Sum_{k>=0} 1/(2*k*(2*k+1)) = A239354 + 1/4 = A188859/2.

From Amiram Eldar, Aug 07 2020: (Start)

Equals Sum_{k>=1} 1/(k*(k+1)*2^k) = Sum_{k>=2} 1/A100381(k).

Equals Sum_{k>=2} (-1)^k * zeta(k)/2^k.

Equals Integral_{x=1..oo} 1/(x^2 + x^3) dx. (End)

Equals log(e/2) = log(A019739) = -log(2/e) = -log(A135002). - Wolfdieter Lang, Mar 04 2022

Equals lim_{n->oo} A024168(n)/n!. - Alois P. Heinz, Jul 08 2022

Equals 1/(4 - 4/(7 - 12/(10 - ... - 2*n*(n-1)/((3*n+1) - ...)))) (an equivalent continued fraction for 1 - log(2) was conjectured by the Ramanujan machine). - Peter Bala, Mar 04 2024

Equals Sum_{k>=1} zeta(2*k)/((2*k + 1)*2^(2*k-1)) (see Finch). - Stefano Spezia, Nov 02 2024

EXAMPLE

0.30685281944005469058276787854...

MAPLE

f:= sum(1/(2*k*(2*k+1)), k=1..infinity):

s:= convert(evalf(f, 140), string):

seq(parse(s[i+1]), i=1..106); # Alois P. Heinz, Jun 17 2014

MATHEMATICA

RealDigits[1-Log[2], 10, 120][[1]] (* Harvey P. Dale, Sep 23 2016 *)

PROG

(PARI) 1-log(2) \\ Charles R Greathouse IV, Jul 14 2014

CROSSREFS

Cf. A002162, A019739, A024168, A100381, A135002, A188859.

Essentially the same digits as A239354.

KEYWORD

nonn,cons,changed

AUTHOR

Franklin T. Adams-Watters, Jun 17 2014

STATUS

approved

A187832

Decimal expansion of integral from 1/2 to 1 of (1-x)/x dx.

+10
6

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Replacing 1/2 with any other number 0 < t < 1, the value of the integral is t - 1 - log(t).

REFERENCES

J.-M. Monier, Cours, Analyse, Tome 4, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 4.3.14 pages 53 and 367.

LINKS

Table of n, a(n) for n=0..104.

FORMULA

Equals log(2) - 1/2 = A002162 - 1/2.

Equals Sum_{k>=1} 1/((2k-1)*(2k)*(2k+1)). - Bruno Berselli, Mar 16 2014

From Amiram Eldar, Jul 28 2020: (Start)

Equals Sum_{k>=0} (-1)^k/(k+3).

Equals Sum_{k>=2} 1/(k * 2^k).

Equals Sum_{k>=2} 1/(4*k^2 - 2*k).

Equals Sum_{k>=2} (zeta(k) - 1)/2^k.

Equals Sum_{k>=1} zeta(2*k + 1)/2^(2*k + 1). (End)

From Bernard Schott, Nov 22 2021: (Start)

Equals Sum_{k>=1} (S(k) - log(2)) when S(k) = Sum_{m=1..k} (-1)^(m+1) / m.

Equals Integral_{x=0..1} x/(1+x)^2 dx. (End)

Equals Sum_{k,m>=1} (-1)^(k+m)/(k+m). - Amiram Eldar, Jun 09 2022

Equals Integral_{x = 0..1} Integral_{y = 0..1} x*y/(x + y)^2 dy dx. - Peter Bala, Dec 12 2022

EXAMPLE

0.193147180559945309417232121458176568075500134360255254120680009493393621969...

MAPLE

(evalf(log(2) - 1/2), 111); # Bernard Schott, Nov 25 2021

MATHEMATICA

RealDigits[Log[2] - 1/2, 10, 111][[1]]

PROG

(PARI) log(2)-1/2 \\ Charles R Greathouse IV, Dec 27 2012

CROSSREFS

Apart from the first digit the same as A002162.

Cf. A239354: Sum_{k>=1} 1/((2k)*(2k+1)*(2k+2)).

KEYWORD

nonn,cons,easy

AUTHOR

Robert G. Wilson v, Dec 27 2012

STATUS

approved

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