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A012245 Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant. 10
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 (list; constant; graph; refs; listen; history; text; internal format)
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].

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

REFERENCES

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.

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

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]

G. Xiao, Contfrac

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

Index entries for characteristic functions

Index entries for continued fractions for constants

FORMULA

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

EXAMPLE

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

0.110001000000000000000001000000000000000000000000000000000000000000000... - Harry J. Smith, May 15 2009

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-Fran├žois 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

CROSSREFS

Cf. A000142, A058304 (continued fraction).

Sequence in context: A179020 A179771 A094875 * A185059 A179776 A089802

Adjacent sequences:  A012242 A012243 A012244 * A012246 A012247 A012248

KEYWORD

nonn,nice,cons,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 15 07:25 EST 2018. Contains 317225 sequences. (Running on oeis4.)