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A012245
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Characteristic function of factorial numbers; also decimal expansion of Liouville's number or Liouville's constant.
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10
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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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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
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FORMULA
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G.f.: Sum_{i>=1} x^Product_{j=1..i} j. - Jon Perry, Mar 31 2004
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EXAMPLE
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a(25) = a(26) = ... = a(119) = 0 because 4! = 24 and 5! = 120.
0.110001000000000000000001000000000000000000000000000000000000000000000... - Harry J. Smith, May 15 2009
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000142, A058304 (continued fraction).
Sequence in context: A179020 A179771 A094875 * A185059 A179776 A089802
Adjacent sequences: A012242 A012243 A012244 * A012246 A012247 A012248
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KEYWORD
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nonn,nice,cons
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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