A002285 - OEIS

%I M3210 N1299 #81 Aug 25 2024 18:47:55

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

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

%U 4,6,4,9,2,0,8,8,7,0,7,7,4,7,2,9,2,2,4,9,4,9,3,3,8,4,3,1,7,4,8,3,1,8,7,0,6

%N Decimal expansion of common logarithm of e.

%C Sometimes also called Briggs's constant after the English mathematician Henry Briggs (1561-1630). - _Martin Renner_, Jan 03 2022

%D 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.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

%H Vincenzo Librandi, <a href="/A002285/b002285.txt">Table of n, a(n) for n = 0..5000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Jonathan Sondow and Eric W. Weisstein, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">MathWorld: Harmonic Number</a>.

%H Horace S. Uhler, <a href="https://doi.org/10.1073%2Fpnas.26.3.205">Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17</a>, Proc. Nat. Acad. Sci. U.S.A. 26, (1940), pp. 205-212.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Henry_Briggs_(mathematician)">Henry Briggs</a>.

%F Equals log_10(e) = 1/log(10) = 1/A002392. - _Eric Desbiaux_, Jun 27 2009

%F Conjecture by Eric Weisstein: Equals lim_{n->oo} b(n)/10^(n-1), for b=A114467 or b=A114468 (i.e., is the limit of the decimal expansion of the number of decimal digits in both the numerator and denominator of the (10^n)th harmonic number). More generally, log_k(e) seems to equal lim_{n->oo} floor(log_k(b(k^n)))/k^(n-1), for b=A001008 or b=A002805 and k >= 2. - _Nathan L. Skirrow_, Feb 12 2023

%e 0.4342944819...

%p evalf[100](1/log(10)); # _Martin Renner_, Jan 03 2022

%t RealDigits[N[1/Log[10], 100]][[1]] (* _Vincenzo Librandi_, Mar 25 2013 *)

%t RealDigits[Log10[E],10,120][[1]] (* _Harvey P. Dale_, Apr 17 2022 *)

%o (PARI) 1/log(10) \\ _Charles R Greathouse IV_, Jan 04 2016

%Y Cf. A001008, A002392, A002805, A114467, A114468.

%K nonn,cons

%O 0,1

%A _N. J. A. Sloane_