

A002392


Decimal expansion of natural logarithm of 10.
(Formerly M0394 N0151)


21



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
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OFFSET

1,1


COMMENTS

10 log 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 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 decimal representation contains only a limited quantity of each digit of a given nonempty set of digits. Actually, such series are all known to be convergent. Here, letting S^{(r)} (r in N}) denote the series of the reciprocals of the positive integers whose decimal representation contains the digit 9 exactly r times, the impressive result obtained is that S^{(r)} tends to 10 log{10} as r tends to infinity!  Jonathan Vos Post, Jul 23 2008


REFERENCES

W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20000
Bakir Farhi, A curious result related to Kempner's series, Jul 22, 2008.
A. J. Kempner, A curious convergent series, Amer. Math. Monthly 23(1914)4850.
Simon Plouffe, log(10) the natural logarithm of 10 to 2000 digits
Simon Plouffe, Plouffe's Inverter, The natural logarithm of 10 to 2000 digits
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). 205212.
Eric Weisstein's World of Mathematics, Natural Logarithm of 10


EXAMPLE

2.302585092994045684017991454684364207601101488628772976033327900967572...


MATHEMATICA

RealDigits[Log[10], 10, 120][[1]] (* Harvey P. Dale, Nov 23 2013 *)


PROG

(PARI) { default(realprecision, 20080); x=log(10); for (n=1, 20000, d=floor(x); x=(xd)*10; write("b002392.txt", n, " ", d)); } \\ Harry J. Smith, Apr 16 2009


CROSSREFS

Cf. A016738 (continued fraction).
Sequence in context: A024307 A219864 A194745 * A002708 A167925 A209927
Adjacent sequences: A002389 A002390 A002391 * A002393 A002394 A002395


KEYWORD

cons,nonn


AUTHOR

N. J. A. Sloane


STATUS

approved



