A188859 Decimal expansion of 2 - log(4).

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

Offset: 0

Author

Charles R Greathouse IV, Apr 12 2011

Keywords

• nonn
• cons
• easy

Comments

Limit as n increases without bound of the probability that n mod m is less than m/2, with m chosen uniformly at random from 1..n. (As usual, 0 <= n mod m < m.)

Examples

			0.61370563888010938116553575708364686384899973127949...
		

Links

• Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
• I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.236.5).
• Index entries for transcendental numbers

Crossrefs

Cf. A016639, A023416, A244009.

Programs

Mathematica

RealDigits[2 - Log[4], 10, 120][[1]]

PARI

vecextract(eval(Vec(Str(2-log(4)))),"3..")

Formula

From Amiram Eldar, Aug 15 2020: (Start)

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

Equals -Integral_{x=0..1} log(1-x^2) dx. (End)

Equals Sum_{k>=1} A023416(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

Equals 1/(1 + 2/(3 + 1^2/(4 + 3^2/(5 + 2^2/(6 + 4^2/(7 + 3^2/(8 + 5^2/(9 + 4^2/(10 + 6^2/(11 + ... + (n-1)^2/((2*n) + (n+1)^2/((2*n+1) + ... )))))))))))). Cf. A016639. - Peter Bala, Mar 04 2024

Equals 1/2 + Sum_{k>=1} 1/(k*(4*k^2-1)^2). - Sean A. Irvine, Apr 06 2025