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%I #17 Apr 30 2022 08:00:41
%S 1,2,3,5,6,9,9,13,14,18,19,20,24,24,23,29,33,36,31,38,41,42,46,50,53,
%T 58,56,57,70,73,77,69,76,76,78,77,80,85,89,101,101,105,106,104,106,
%U 112,115,124,113,126,124,124,130,144,144,148,140,149,141,151,157,158,172
%N Number of terms in the simple continued fraction for Sum_{k=1..n} 1/k!.
%H Amiram Eldar, <a href="/A069880/b069880.txt">Table of n, a(n) for n = 1..10000</a>
%H Amiram Eldar, <a href="/A069880/a069880.jpg">Plot of a(n)/(n * log(log(n))) for n = 1..10000</a>.
%F Does lim_{n->infinity} a(n)/(n * log(log(n))) = C = 2.XXX...?
%e For n=4, Sum_{k=1..n} 1/k! = 1/1! + 1/2! + 1/3! + 1/4! = 1/1 + 1/2 + 1/6 + 1/24 = 41/24 = 1 + 1/(1 + 1/(2 + 1/(2 + 1/3))) = CF[1;1,2,2,1], so a(4) = 5.
%t lcf[f_] := Length[ContinuedFraction[f]]; lcf /@ Accumulate[Table[1/k!, {k, 1, 100}]] (* _Amiram Eldar_, Apr 30 2022 *)
%Y Cf. A055573, A061354, A061355, A070267.
%K easy,nonn
%O 1,2
%A _Benoit Cloitre_, May 04 2002