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

1,4

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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A053510 (decimal expansion).

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

Sequence in context: A204257 A074976 A329873 * A054081 A164585 A200996

Adjacent sequences: A068445 A068446 A068447 * A068449 A068450 A068451

Keyword

nonn

Author

Benoit Cloitre, Mar 10 2002

Extensions

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

Status

approved