A068448 Factorial expansion of log(Pi) = Sum_{n>0} a(n)/n! with a(n) as large as possible.

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, 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, 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

Offset: 1

Author

Benoit Cloitre, Mar 10 2002

Keywords

• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Wikipedia, Factorial number system: Fractional values
• Index to sequences related to factorial base representation of noninteger constants.

Crossrefs

Cf. A053510 (decimal expansion).

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

Programs

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

Mathematica

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 *)

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

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

Extensions

Name edited by M. F. Hasler, Dec 04 2018