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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• cons

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].

Examples

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

References

• Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 89.
• John H. Conway and 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.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.

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.

Crossrefs

Cf. A000142, A058304 (continued fraction).

Programs

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 *)

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

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