A244009 Decimal expansion of 1 - log(2).

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

Offset: 0

Author

Franklin T. Adams-Watters, Jun 17 2014

Keywords

• nonn
• cons

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

Examples

			0.30685281944005469058276787854...
		

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

• 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

Crossrefs

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

Essentially the same digits as A239354.

Programs

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 *)

PARI

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

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