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A016627 Decimal expansion of log(4). 35
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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)
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.
LINKS
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.
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.
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. - 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_{n >= 1} 1/(n*P(n, 5/3)*P(n-1, 5/3)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(4) = 1.386294361119890618(66...), correct to 18 decimal places. - Peter Bala, Mar 18 2024
EXAMPLE
1.38629436111989061883446424291635313615100026872051050824136...
MATHEMATICA
RealDigits[Log@ 4, 10, 111][[1]] (* Robert G. Wilson v, Aug 31 2014 *)
PROG
(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
CROSSREFS
Cf. A016732 (continued fraction).
Cf. A002162 (half), A133362 (reciprocal).
Sequence in context: A021263 A246727 A081803 * A175184 A019604 A336079
KEYWORD
nonn,cons
AUTHOR
STATUS
approved

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Last modified July 12 22:19 EDT 2024. Contains 374257 sequences. (Running on oeis4.)