A012245 Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant.

1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

1,1

Comments

Read as decimal fraction 1100010... in any base > 1 (arbitrary decimal point) Liouville's numbers are transcendental; read as a continued fraction it is also transcendental [G. H. Hardy and E. M. Wright, Th. 192].

References

John H. Conway & Richard K. Guy, The Book of Numbers, pp. 239-241 (1996).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 162.

T. W. Koerner, Fourier Analysis, Camb. Univ. Press 1988, p. 177.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 58.

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

J. Liouville, Communication, C. R. Acad. Sci. Paris 18, 883-885 and 993-995, 1844. [Pages 993-995 do not seem right]

J. Liouville, Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques, Journal de Mathématiques Pures et Appliquées 16, pp. 133-142, 1851.

Diego Marques and Carlos Gustavo Moreira, On variations of the Liouville constant which are also Liouville numbers, Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 3 (2016), 39-40.

Michael Penn, One of the first transcendental numbers -- Liouville's Constant, YouTube video, 2022.

Burkard Polster, Liouville's number, the easiest transcendental and its clones, Mathologer video (2017).

Eric Weisstein's World of Mathematics, Liouville's Constant

G. Xiao, Contfrac

Index entries for characteristic functions

Index entries for continued fractions for constants

Index entries for transcendental numbers

Formula

G.f.: Sum_{i>=1} x^Product_{j=1..i} j. - Jon Perry, Mar 31 2004

a(A000142(n)) = 1; a(A063992(n)) = 0. - Reinhard Zumkeller, Oct 11 2008

Example

a(25) = a(26) = ... = a(119) = 0 because 4! = 24 and 5! = 120.
0.110001000000000000000001000000000000000000000000000000000000000000000....

Mathematica

With[{nn=5}, ReplacePart[Table[0, {nn!}], Table[{n!}, {n, nn}]->1]] (* Harvey P. Dale, Jul 22 2012 *)
RealDigits[ Sum[1/10^n!, {n, 5}], 10, 105][[1]] (* Robert G. Wilson v, Aug 03 2018 *)
CoefficientList[1/x Sum[x^k!, {k, 1, 5}], x] (* Jean-François Alcover, Nov 02 2018 *)

Prog

(PARI) default(realprecision, 20080); x=10*suminf(n=1, 1.0/10^n!) + 1/10^20040; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b012245.txt", n, " ", d)); \\ Harry J. Smith, May 15 2009
(Python)
from itertools import count
def A012245(n):
    c = 1
    for i in count(1):
        if (c:=c*i) >= n:
            return int(c==n) # Chai Wah Wu, Jan 11 2023

Crossrefs

Cf. A000142, A058304 (continued fraction).

Sequence in context: A194667 A094875 A359456 * A256436 A253903 A255849

Adjacent sequences: A012242 A012243 A012244 * A012246 A012247 A012248

Keyword

nonn,nice,cons

Author

N. J. A. Sloane

Status

approved