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A322333 Factorial expansion of log(5) = Sum_{n>=1} a(n)/n!.

%I A322333 #17 May 24 2022 00:11:08
%S A322333 1,1,0,2,3,0,5,4,4,8,3,3,2,0,7,8,0,7,11,1,18,16,3,10,16,21,17,13,20,
%T A322333 12,16,8,27,24,28,12,9,34,21,3,9,8,41,42,35,31,4,4,37,38,9,20,10,31,
%U A322333 24,34,44,21,16,19,24,4,22,22,47,8,28,26,32,22,28,56,44,16,61,38,3,25,52,35,73,55,8,42,25,21,62,61,7,89,5,74,89,57,33,60,13,75,95,66
%N A322333 Factorial expansion of log(5) = Sum_{n>=1} a(n)/n!.
%H A322333 G. C. Greubel, <a href="/A322333/b322333.txt">Table of n, a(n) for n = 1..10000</a>
%H A322333 <a href="https://oeis.org/index/Fa#facbase">Index entries for factorial base representation of noninteger constants</a>
%e A322333 log(5) = 1 + 1/2! + 0/3! + 2/4! + 3/5! + 0/6! + 5/7! + 4/8! + ...
%t A322333 With[{b = Log[5]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]
%o A322333 (PARI) default(realprecision, 250); b = log(5); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))
%o A322333 (Magma) SetDefaultRealField(RealField(250));  [Floor(Log(5))] cat [Floor(Factorial(n)*Log(5)) - n*Floor(Factorial((n-1))*Log(5)) : n in [2..80]];
%o A322333 (Sage)
%o A322333 def a(n):
%o A322333     if (n==1): return floor(log(5))
%o A322333     else: return expand(floor(factorial(n)*log(5)) - n*floor(factorial(n-1)*log(5)))
%o A322333 [a(n) for n in (1..80)]
%Y A322333 Cf. A016628 (decimal expansion), A016733 (continued fraction).
%Y A322333 Cf. A067882 (log(2)), A322334 (log(3)), A068460 (log(7)), A068461 (log(11)).
%K A322333 nonn
%O A322333 1,4
%A A322333 _G. C. Greubel_, Dec 03 2018