A016627 Decimal expansion of log(4).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

This constant (negated) is the 1-dimensional analog of Madelung's constant. - Jean-François Alcover, May 20 2014

This constant is the sum over the reciprocals of the hexagonal numbers A000384(n), n >= 1. See the Downey et al. link, and the formula by Robert G. Wilson v below. - Wolfdieter Lang, Sep 12 2016

log(4) - 1 is the mean ratio between the smaller length and the larger length of the two parts of a stick that is being broken at a point that is uniformly chosen at random (Mosteller, 1965). - Amiram Eldar, Jul 25 2020

From Bernard Schott, Sep 11 2020: (Start)

This constant was the subject of the problem B5 during the 42nd Putnam competition in 1981 (see formula Sep 11 2020 and Putnam link).

Jeffrey Shallit generalizes this result obtained for base 2 to any base b (see Amer. Math. Month. link): Sum_{k>=1} digsum(k)_b / (k*(k+1)) = (b/(b-1)) * log(b) where digsum(k)_b is the sum of the digits of k when expressed in base b (for base 10 see A334388). (End)

Examples

			1.38629436111989061883446424291635313615100026872051050824136...
		

References

• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
• Frederick Mosteller, Fifty challenging problems of probability, Dover, New York, 1965. See problem 42, pp. 10 and 63.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:13:8 at page 23.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
• I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.236.1 and 0.236.6).
• D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly, Vol. 92, No. 7 (1985) pp. 449-457.
• Allon G. Percus, Gabriel Istrate, Bruno Goncalves, Robert Z. Sumi, and Stefan Boettcher, The Peculiar Phase Structure of Random Graph Bisection, arXiv:0808.1549 [cond-mat.stat-mech], 2008.
• H.-J. Seiffert, Problem B-771, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 32, No. 4 (1994), p. 374; More Sums, Solution to Problem B-771 by Don Redmond, ibid., Vol. 33, No. 5 (1995), pp. 470-471.
• J. O. Shallit, Solutions of Advanced Problems, 6450, The American Mathematical Monthly, Vol. 92, No. 7, Aug.-Sep., 1985, pp. 513-514; DOI: 10.2307/2322523.
• 42nd Putnam Competition, Problem B5, 1981.
• Index entries for sequences related to Olympiads and other Mathematical competitions.
• Index entries for transcendental numbers.

Crossrefs

Cf. A016732 (continued fraction).

Cf. A002162 (half), A133362 (reciprocal).

Cf. A000384, A001787, A002378, A007531, A066066, A069072, A191907.

Cf. A000045, A000120, A334388.

Programs

Mathematica

RealDigits[Log@ 4, 10, 111][[1]] (* Robert G. Wilson v, Aug 31 2014 *)

PARI

default(realprecision, 20080); x=log(4); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016627.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009, corrected May 19 2009

PARI

A016627_vec(N)=digits(floor(log(precision(4.,N))*10^(N-1))) \\ Or: default(realprecision,N);digits(log(4)\.1^N) \\ M. F. Hasler, Oct 20 2013

Formula

log(4) = Sum_{k >= 1} H(k)/2^k where H(k) is the k-th harmonic number. - Benoit Cloitre, Jun 15 2003

Equals 1 - Sum_{k >= 1} (-1)^k/A002378(k) = 1 + 2*Sum_{k >= 0} 1/A069072(k) = 5/4 - Sum_{k >= 1} (-1)^k/A007531(k+2). - R. J. Mathar, Jan 23 2009

Equals 2*A002162 = Sum_{n >= 1} binomial(2*n, n)/(n*4^n) [D. H. Lehmer, Am. Math. Monthly 92 (1985) 449 and Jolley eq. 262]. - R. J. Mathar, Mar 04 2009

log(4) = Sum_{k >= 1} A191907(4, k)/k, (conjecture). - Mats Granvik, Jun 19 2011

log(4) = lim_{n -> infinity} A066066(n)/n. - M. F. Hasler, Oct 20 2013

Equals Sum_{k >= 1} 1/( 2*k^2 - k ). - Robert G. Wilson v, Aug 31 2014

Equals gamma(0, 1/2) - gamma(0, 1) = -(EulerGamma + polygamma(0, 1/2)), where gamma(n,x) denotes the generalized Stieltjes constants, see A020759. - Peter Luschny, May 16 2018

From Amiram Eldar, Jul 25 2020: (Start)

Equals Sum_{k>=1} (3/4)^k/k.

Equals Sum_{k>=1} 1/(k*2^(k-1)) = Sum_{k>=1} 1/A001787(k).

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

Equals Sum_{k>=1} A000120(k) / (k*(k+1)). - Bernard Schott, Sep 11 2020

Equals 1 + Sum_{k>=1} zeta(2*k+1)/4^k. - Amiram Eldar, May 27 2021

Equals Sum_{k>=1} (2*k+1)*Fibonacci(k)/(k*(k+1)*2^k) (Seiffert, 1994). - Amiram Eldar, Jan 15 2022

Continued fraction: log(4) = 1 + 1/(2 + (1*2)/(2 + (2*3)/(2 + (3*4)/(2 + (4*5)/(2 + ... ))))) due to Euler. - Peter Bala, Mar 05 2024

log(4) = 2*Sum_{k>=1} 1/(k*P(k, 5/3)*P(k-1, 5/3)), where P(k, x) denotes the k-th Legendre polynomial. The first 20 terms of the series gives log(4) correct to 18 decimal places. - Peter Bala, Mar 18 2024

Equals Sum_{k>=1} (2*k - 1)!!/(k*(2*k)!!) [Ross] (see Spanier at p. 23). - Stefano Spezia, Dec 27 2024

Equals 1 + Sum_{k>=1} 1/(k*(4*k^2-1)). - Sean A. Irvine, Apr 05 2025

Equals Sum_{k>=1} (12*k^2-1)/(k*(4*k^2-1)^2). - Sean A. Irvine, Apr 06 2025

Equals Integral_{x=0..1} arctanh(sqrt(x))/sqrt(x) dx. - Kritsada Moomuang, Jun 06 2025

From Kritsada Moomuang, Jun 18 2025: (Start)

Equals Integral_{x=0..1} (x^(n - 1)*(x^(3*n) - 1))/log(x) dx, for n > 0.

Equals Integral_{x=0..Pi} sin(x)/(1 + abs(cos(x))) dx. (End)