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Engel expansion of log(Pi) = 1.14473... .
2

%I #26 Apr 13 2018 11:24:06

%S 1,7,77,107,150,167,7091,27852,31790,34069,327724,416403,4669290,

%T 20206510,2218014305,4524826037,4576058224,5496581959,15869888136,

%U 91151928112,104430320239,202761572952,218933128153,937032410920,1044739832405,15262515810234

%N Engel expansion of log(Pi) = 1.14473... .

%C Cf. A006784 for definition of Engel expansion.

%D F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

%H G. C. Greubel, <a href="/A059195/b059195.txt">Table of n, a(n) for n = 1..1000</a>

%H 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 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 <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>

%t EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@

%t NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];

%t EngelExp[N[Log[Pi], 7!], 100] (* Modified by _G. C. Greubel_, Dec 28 2016 *)

%Y Cf. A053510.

%K nonn,easy,nice

%O 1,2

%A _Mitch Harris_