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Decimal expansion of log(16) = 4*log(2).
9

%I #58 Jun 10 2024 07:13:35

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

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

%U 7,4,4,8,7,8,7,8,7,7,8,8,6,2,4,2,3,4,5,3,3,0,7,9,8,5,6,7,4,7,5

%N Decimal expansion of log(16) = 4*log(2).

%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.

%H Harry J. Smith, <a href="/A016639/b016639.txt">Table of n, a(n) for n = 1..20000</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 Peter Bala, <a href="/A016639/a016639.pdf"> A continued fraction expansion for the constant log(16) </a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MadelungConstants.html">Madelung Constants</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals 4*A002162.

%F Equals 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

%F Equals -2 + Sum_{k>=1} H(k)*(k+1)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - _Amiram Eldar_, May 28 2021

%F Equals 1 + Limit_{n -> infinity} (1/n)*Sum_{k = 1..n} (2*n + k)/(2*n - k) = 2*( 1 + Limit_{n -> infinity} (1/n)*Sum_{k = 1..n} (n - k)/(n + k) ). - _Peter Bala_, Oct 10 2021

%F Equals 2 + 1/(1 + 1/(3 + 2/(4 + 6/(5 + 6/(6 + 12/(7 + 12/(8 + ... + n*(n-1)/(2*n-1 + n*(n-1)/(2*n + ...))))))))). Cf. A188859. - _Peter Bala_, Mar 04 2024

%e 2.77258872223978123766892848583270627230200053744102101648272... - _Harry J. Smith_, May 17 2009

%t RealDigits[Log[16], 10, 120][[1]] (* _Harvey P. Dale_, Jun 12 2012 *)

%o (PARI) default(realprecision, 20080); x=log(16); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016639.txt", n, " ", d)); \\ _Harry J. Smith_, May 17 2009, corrected May 19 2009

%o (Magma) Log(16); // _Vincenzo Librandi_, Feb 20 2015

%Y Equals 4*A002162.

%Y Equals (4/5)*A016655.

%Y Equals A303658 + 2.

%Y Cf. A016444 (continued fraction).

%Y Cf. A001008, A002805, A188859.

%K nonn,cons

%O 1,1

%A _N. J. A. Sloane_