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A012245 Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant. 12
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].
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
J. Liouville, Communication, C. R. Acad. Sci. Paris 18, 883-885 and 993-995, 1844. [Pages 993-995 do not seem right]
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.
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
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
KEYWORD
nonn,nice,cons
AUTHOR
STATUS
approved

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Last modified May 20 05:07 EDT 2024. Contains 372703 sequences. (Running on oeis4.)