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Search: id:A012245
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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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+0
6
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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; graph; listen)
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OFFSET
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1,1
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COMMENT
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Read as decimal fraction 1100010... in any base > 1 (arbitrary decimal point) Lioville'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. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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.
J. Liouville, C. R. Acad. Sci. Paris 18, 883-885 and 993-995, 1844.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 58.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=1,...,20000
Index entries for characteristic functions
G. Xiao, Contfrac
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for continued fractions for constants
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FORMULA
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G.f.: sum(i=1, oo, x^product(j=1, i, j)) - Jon Perry (perry(AT)globalnet.co.uk), 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... [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 15 2009]
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PROGRAM
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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)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 15 2009]
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CROSSREFS
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Cf. A000142.
Cf. A058304 Continued fraction. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 15 2009]
Sequence in context: A113052 A117964 A094875 this_sequence A089801 A089802 A143064
Adjacent sequences: A012242 A012243 A012244 this_sequence A012246 A012247 A012248
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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