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Search: id:A002392
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| A002392 |
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Decimal expansion of natural logarithm of 10.
(Formerly M0394 N0151)
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+0
8
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| 2, 3, 0, 2, 5, 8, 5, 0, 9, 2, 9, 9, 4, 0, 4, 5, 6, 8, 4, 0, 1, 7, 9, 9, 1, 4, 5, 4, 6, 8, 4, 3, 6, 4, 2, 0, 7, 6, 0, 1, 1, 0, 1, 4, 8, 8, 6, 2, 8, 7, 7, 2, 9, 7, 6, 0, 3, 3, 3, 2, 7, 9, 0, 0, 9, 6, 7, 5, 7
(list; cons; graph; listen)
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OFFSET
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1,1
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COMMENT
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10 ln 10 ~ 23.0258509299... appears in Bakir Farhi's paper. Abstract: It is well known since A. J. Kempner's work that the series of the reciprocals of the positive integers whose the decimal representation does not contain any digit 9, is convergent. This result was extended by F. Irwin and others to deal with the series of the reciprocals of the positive integers whose the decimal representation contains only a limited quantity of each digit of a given nonempty set of digits. Actually, such series are known to be all convergent. Here, letting S^{(r)} (r in N}) denote the series of the reciprocal of the positive integers whose the decimal representation contains the digit 9 exactly r times, the impressive obtained result is that S^{(r)} tends to 10 log{10} as r tends to infinity! - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 23 2008
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
A. J. Kempner, A curious convergent series, Amer. Math. Monthly 23(1914)48-50.
W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. 2.
Uhler, Horace S.; Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17. Proc. Nat. Acad. Sci. U. S. A. 26, (1940). 205-212.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=1,...,20000
S. Plouffe, log(10) the natural logatithm of 10 to 2000 digits
S. Plouffe, Plouffe's Inverter, The natural logarithm of 10 to 2000 digits
Eric Weisstein's World of Mathematics, Natural Logarithm of 10
Bakir Farhi, A curious result related to Kempner's series, Jul 22, 2008.
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EXAMPLE
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2.302585092994045684017991454684364207601101488628772976033327900967572... [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 16 2009]
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PROGRAM
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(PARI) { default(realprecision, 20080); x=log(10); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002392.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 16 2009]
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CROSSREFS
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Cf. A016738 Continued fraction. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 16 2009]
Sequence in context: A103180 A126045 A024307 this_sequence A002708 A059283 A160202
Adjacent sequences: A002389 A002390 A002391 this_sequence A002393 A002394 A002395
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KEYWORD
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cons,nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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