cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A244009 Decimal expansion of 1 - log(2).

Original entry on oeis.org

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

Views

Author

Franklin T. Adams-Watters, Jun 17 2014

Keywords

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

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

A187832 Decimal expansion of integral from 1/2 to 1 of (1-x)/x dx.

Original entry on oeis.org

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
Offset: 0

Views

Author

Robert G. Wilson v, Dec 27 2012

Keywords

Comments

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

Examples

			0.193147180559945309417232121458176568075500134360255254120680009493393621969...

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

Crossrefs

Apart from the first digit the same as A002162.
Cf. A239354: Sum_{k>=1} 1/((2k)*(2k+1)*(2k+2)).

Programs

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
Showing 1-2 of 2 results.

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