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A059180 Engel expansion of log(2).

%I A059180 #36 Feb 16 2025 08:32:43
%S A059180 2,3,7,9,104,510,1413,2386,40447,87110,124975,1565154,1766158,2440919,
%T A059180 2637001,9192874,24998746,73973182,88828340,432049320,470421590,
%U A059180 477600793,3313014448,4571423959,28839435286,40818751774
%N A059180 Engel expansion of log(2).
%C A059180 See A006784 for the definition of Engel expansion.
%D A059180 F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.
%H A059180 G. C. Greubel and T. D. Noe, <a href="/A059180/b059180.txt">Table of n, a(n) for n = 1..1000</a>[Terms 1 to 300 computed by T. D. Noe; Terms 301 to 1000 computed by G. C. Greubel, Dec 27 2016]
%H A059180 F. Engel, <a href="/A006784/a006784.pdf">Entwicklung der Zahlen nach Stammbruechen</a>, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
%H A059180 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H A059180 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NaturalLogarithmof2.html">Natural Logarithm of 2</a>
%H A059180 P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
%H A059180 <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%t A059180 EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
%t A059180 NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
%t A059180 EngelExp[N[Log[2], 7!], 100] (* Modified by _G. C. Greubel_, Dec 27 2016 *)
%Y A059180 Cf. A002162 (decimal expansion of the natural logarithm of 2).
%K A059180 nonn,easy,nice
%O A059180 1,1
%A A059180 _Mitch Harris_