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%I #11 Jan 19 2019 04:05:14
%S 1,1,1,2,3,4,2,4,7,5,6,5,12,1,12,9,0,7,4,14,10,17,2,14,23,4,2,2,16,2,
%T 10,18,23,26,26,26,24,1,17,26,18,12,0,15,42,34,39,33,20,18,40,43,12,
%U 47,51,10,50,35,14,23,16,1,55,41,34,29,14,41,35,60,53,45,61,35,49,73,13,13,57,59
%N Factorial expansion of exp(gamma) = Sum_{n>=1} a(n)/n! with a(n) as large as possible.
%C Gamma is the Euler-Mascheroni constant (A001620).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicExpansion.html">Harmonic Expansion</a>
%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Fa#facbase">Index entries for factorial base representation</a>
%F Sum_{n>=1} a(n)/n! = exp(gamma) = A073004.
%e exp(gamma) = 1 + 1/2! + 1/3! + 2/4! + 3/5! + 4/6! + 2/7! + 4/8! + ...
%p Digits:=200: a:=n->`if`(n=1,floor(exp(gamma)),floor(factorial(n)*exp(gamma))-n*floor(factorial(n-1)*exp(gamma))): seq(a(n),n=1..100); # _Muniru A Asiru_, Dec 20 2018
%t With[{b = Exp[EulerGamma]}, Table[If[n==1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]
%o (PARI) default(realprecision, 250); b = exp(Euler); for(n=1, 80, print1( if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))
%Y Cf. A073004 (decimal expansion), A094644 (continued fraction), A001620 (Euler-Mascheroni constant).
%K nonn
%O 1,4
%A _Tristan Cam_, Dec 20 2018
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