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Factorial expansion of log(Pi) = Sum_{n>0} a(n)/n! with a(n) as large as possible.
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%I #18 Sep 08 2022 08:45:05

%S 1,0,0,3,2,2,1,3,4,5,8,10,11,7,13,13,3,14,11,16,6,9,3,14,0,16,22,9,8,

%T 26,5,18,6,3,13,31,4,27,25,5,12,1,17,31,2,4,16,17,39,15,15,25,52,40,

%U 50,3,27,32,54,18,55,10,29,62,38,4,17,53,13,24,22,40,23,11,74,18,74,31,8

%N Factorial expansion of log(Pi) = Sum_{n>0} a(n)/n! with a(n) as large as possible.

%C If a(n) is not required to be as large as possible, it isn't well defined: it can be decreased by any amount x without changing the value of the sum, if x*(n+1) is added to a(n+1), which in turn can be decreased by any arbitrary amount etc. - _M. F. Hasler_, Dec 04 2018

%H G. C. Greubel, <a href="/A068448/b068448.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system#Fractional_values">Factorial number system: Fractional values</a>

%H <a href="/index/Fa#facbase">Index to sequences related to factorial base representation of noninteger constants</a>.

%t Table[If[n == 1, Floor[Log[Pi]], Floor[n!*Log[Pi]] - n*Floor[(n - 1)!*Log[Pi]]], {n,1,50}] (* _G. C. Greubel_, Mar 21 2018 *)

%o (PARI) for(n=1,30, print1(if(n==1, floor(log(Pi)), floor(n!*log(Pi)) - n*floor((n-1)!*log(Pi))), ", ")) \\ _G. C. Greubel_, Mar 21 2018

%o (PARI) A068448_vec(N=90,c=log(precision(Pi,N*log(N/2.4)\/2.3)))=vector(N,n,if(n>1,c=c%1*n,c)\1) \\ N*log(N/2.4)\/2.3 ~ logint(N!,10) but uses much less memory when N is big. - _M. F. Hasler_, Nov 28 2018

%o (Magma) R:= RealField(); [Floor(Log(Pi(R)))] cat [Floor(Factorial(n)*Log(Pi(R))) - n*Floor(Factorial((n-1))*Log(Pi(R))) : n in [2..30]]; // _G. C. Greubel_, Mar 21 2018

%Y Cf. A053510 (decimal expansion).

%Y Similar expansions: A068450 (sqrt(Pi)), A075874 (Pi), A007514 (a different variant for Pi).

%K nonn

%O 1,4

%A _Benoit Cloitre_, Mar 10 2002

%E Name edited by _M. F. Hasler_, Dec 04 2018