A016655 Decimal expansion of log(32) = 5*log(2).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

The function exp(x) has its maximum curvature where x = -(1/10)*5*log(2) = -log(2)/2 = 0.34657... - Dimitri Papadopoulos, Oct 27 2022

This maximum curvature occurs at the point with coordinates (x_M = -log(2)/2 = -(this constant)/10; y_M = sqrt(2)/2 = A010503) and is equal to 2*sqrt(3)/9 = A212886. - Bernard Schott, Dec 23 2022

Examples

			3.465735902799726547086160607290882840377500671801276270603400047466968...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• O. Espinosa, V. H. Moll, On some integrals involving the Hurwitz Zeta function: Part 1, Raman. J. 6 (2002) 159-188, eq. (5.7)
• Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021.
• R. S. Melham and A. G. Shannon, Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
• Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), chap 7.1
• H. M. Srivastava, M. L. Glasser, V. S. Adamchik, Some definite integrals associated with the Riemann Zeta Function, Z. Anal. Anw. 19 (3) (2000) 831-846
• Index to sequences related to curves.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000045, A000032, A060851, A195909, A195913, A195697, A016460 (continued fraction).

Cf. A010503, A212886.

Programs

Magma

[5*Log(2)]; // Vincenzo Librandi, Jan 02 2016

Mathematica

RealDigits[5 N [Log[2], 100]] [[1]] (* Vincenzo Librandi, Jan 02 2016 *)

PARI

log(32) \\ Charles R Greathouse IV, Jan 24 2012

Formula

log(2)/2 = (1 - 1/2 - 1/4) + (1/3 - 1/6 - 1/8) + (1/5 - 1/10 - 1/12) + ... [Jolley, Summation of Series, Dover (1961) eq (73)]

Equals 10*log(2)/2 = 5*log(2) = 5*A002162, so 10*(1/2 - 1/4 + 1/6 - 1/8 + 1/10 - ... + (-1)^(k+1)/(2*k) + ...) = log(32). - Eric Desbiaux, Nov 26 2008

-log(2)/2 = lim_{n->oo} ((Sum_{k=2..n} arctanh(1/k)) - log(n)). - Jean-François Alcover, Aug 07 2014, after Steven Finch

Equals log(sqrt(2)) with offset 0. - Michel Marcus, Feb 19 2017

Equals (5/4)*Sum_{k=1..4} (-1)^(k+1) gamma(0, k/4) where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018

From Amiram Eldar, Jun 29 2020: (Start)

log(2)/2 = arctanh(1/3) = arcsinh(1/sqrt(8)).

log(2)/2 = Integral_{x=0..Pi/4} tan(x) dx.

log(2)/2 = Sum_{k>=0} (-1)^k/(2*k+2).

log(2)/2 = Sum_{k>=1} 1/A060851(k). (End)

log(2)/2 = Sum_{k>=1} (-1)^(k+1) * arctanh(Lucas(2*k+3)/Fibonacci(2*k+3)^2) (Melham and Shannon, 1995). - Amiram Eldar, Jan 15 2022

Equals 10 * Integral_{1..oo} dx/(x*(1+x^2)). [Nahin] - R. J. Mathar, May 22 2024

Equals -10*Integral_{q=0..1} q*log(sin(Pi*q))dq. [Espinosa] - R. J. Mathar, Aug 13 2024

log(2)/2 = Sum_{k>=2} (-1)^(k) * arccoth(k). - Antonio Graciá Llorente, Sep 16 2024

-0.34657359... = Sum_{k>=0} zeta(2k)/((2k+1)*2^(2k)), [Srivastava (2.20)] - R. J. Mathar, Feb 12 2020

Equals 10*Integral_{x=0..1} Ei((1 + sqrt(2))*log(x)) - li(x) dx, where Ei is the exponential integral and li is the logarithmic integral. - Kritsada Moomuang, Jun 06 2025