A082838 -id:A082838 - cp's OEIS frontend

A007095 Numbers in base 9.

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84

Offset: 0

N. J. A. Sloane and Robert G. Wilson v

Also numbers without 9 as a digit.

Complement of A011539: A102683(a(n)) = 0; A068505(a(n)) != a(n)). - Reinhard Zumkeller, Dec 29 2011

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
M. F. Hasler, Numbers avoiding certain digits, OEIS wiki, Jan 12 2020
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992
Index entries for 10-automatic sequences.

Cf. A000042 (base 1), A007088 (base 2), A007089 (base 3), A007090 (base 4), A007091 (base 5), A007092 (base 6), A007093 (base 7), A007094 (base 8); A057104, A037479.
Cf. A052382 (without 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8).
Cf. A031076, A031087.
Cf. A082838.

a007095 = f . subtract 1 where
   f 0 = 0
   f v = 10 * f w + r   where (w, r) = divMod v 9
-- Reinhard Zumkeller, Oct 07 2014, Dec 29 2011

[ n: n in [0..74] | not 9 in Intseq(n) ];  // Bruno Berselli, May 28 2011

A007095 := proc(n) local l: if(n=0)then return 0: fi: l:=convert(n,base,9): return op(convert(l,base,10,10^nops(l))): end: seq(A007095(n),n=0..67); # Nathaniel Johnston, May 06 2011

Table[ FromDigits[ IntegerDigits[n, 9]], {n, 0, 75}]

a(n)=if(n<1,0,if(n%9,a(n-1)+1,10*a(n/9)))

A007095(n)=fromdigits(digits(n, 9)) \\ Michel Marcus, Dec 29 2018

# and others: see OEIS Wiki page (cf. LINKS).

from gmpy2 import digits
def A007095(n): return int(digits(n,9)) # Chai Wah Wu, May 06 2025

seq 0 1000 | grep -v 9; # Joerg Arndt, May 29 2011

a(0) = 0, a(n) = 10*a(n/9) if n==0 (mod 9), a(n) = a(n-1)+1 otherwise. - Benoit Cloitre, Dec 22 2002

Sum_{n>1} 1/a(n) = A082838 = 22.92067... (Kempner series). - Bernard Schott, Dec 29 2018; edited by M. F. Hasler, Jan 13 2020

A082830 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 1 in base 10} 1/k.

Original entry on oeis.org

1, 6, 1, 7, 6, 9, 6, 9, 5, 2, 8, 1, 2, 3, 4, 4, 4, 2, 6, 6, 5, 7, 9, 6, 0, 3, 8, 8, 0, 3, 6, 4, 0, 0, 9, 3, 0, 5, 5, 6, 7, 2, 1, 9, 7, 9, 0, 7, 6, 3, 1, 3, 3, 8, 6, 4, 5, 1, 6, 9, 0, 6, 4, 9, 0, 8, 3, 6, 3, 6, 2, 9, 8, 8, 9, 9, 9, 9, 9, 6, 4, 5, 6, 3, 8, 8, 8, 6, 2, 1, 4, 6, 2, 6, 6, 8, 5, 0, 2, 8, 6, 2, 9, 7, 7

Offset: 2

Robert G. Wilson v, Apr 14 2003

Such sums are called Kempner series, see A082839 (the analog for digit 0) for more information. - M. F. Hasler, Jan 13 2020

			16.17696952812344426657...

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. [From _Robert G. Wilson v_, Jun 01 2009]
Eric Weisstein's World of Mathematics,, Kempner Series. [From _R. J. Mathar_, Aug 07 2010]
Wikipedia, Kempner series.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series. [From _Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052383 (numbers without '1'), A011531 (numbers with '1').

Cf. A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 2, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

Equals Sum_{k in A052383\{0}} 1/k, where A052383 = numbers with no digit 1. Those which have a digit 1 (A011531) are omitted in the harmonic sum, and they have asymptotic density 1: almost all terms are omitted from the sum. - M. F. Hasler, Jan 15 2020

More terms from Robert G. Wilson v, Jun 01 2009

A082839 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 0 in base 10} 1/k.

Original entry on oeis.org

2, 3, 1, 0, 3, 4, 4, 7, 9, 0, 9, 4, 2, 0, 5, 4, 1, 6, 1, 6, 0, 3, 4, 0, 5, 4, 0, 4, 3, 3, 2, 5, 5, 9, 8, 1, 3, 8, 3, 0, 2, 8, 0, 0, 0, 0, 5, 2, 8, 2, 1, 4, 1, 8, 8, 6, 7, 2, 3, 0, 9, 4, 7, 7, 2, 7, 3, 8, 7, 5, 0, 7, 9, 6, 0, 6, 1, 4, 1, 9, 4, 2, 6, 3, 5, 9, 2, 0, 1, 9, 1, 0, 5, 2, 6, 1, 3, 9, 3, 3, 8, 6, 5, 2, 1

Offset: 2

Robert G. Wilson v, Apr 14 2003

"The most novel culling of the terms of the harmonic series has to be due to A. J. Kempner, who in 1914 considered what would happen if all terms are removed from it which have a particular digit appearing in their denominators. For example, if we choose the digits 7, we would exclude the terms with denominators such as 7, 27, 173, 33779, etc. There are 10 such series, each resulting from the removal of one of the digits 0, 1, 2, ..., 9 and the first question which naturally arises is just what percentage of the terms of the series are we removing by the process?"
"The sum of the reciprocals, 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... [A002387] is unbounded. By taking sufficiently many terms, it can be made as large as one pleases. However, if the reciprocals of all numbers that when written in base 10 contain at least one 0 are omitted, then the sum has the limit, 23.10345... [Boas and Wrench, AMM v78]." - Wells.
Sums of this type are now called Kempner series, cf. LINKS. Convergence of the series is not more surprising than, and related to the fact that almost all numbers are pandigital (these have asymptotic density 1), i.e., "almost no number lacks any digit": Only a fraction of (9/10)^(L-1) of the L-digit numbers don't have a digit 0. Using L-1 = [log_10 k] ~ log_10 k, this density becomes 0.9^(L-1) ~ k^(log_10 0.9) ~ 1/k^0.046. If we multiply the generic term 1/k with this density, we have a converging series with value zeta(1 - log_10 0.9) ~ 22.4. More generally, almost all numbers contain any given substring of digits, e.g., 314159, and the sum over 1/k becomes convergent even if we omit just the terms having 314159 somewhere in their digits. - M. F. Hasler, Jan 13 2020

			23.10344790942054161603...

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, p. 258.
Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, p. 34.
David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing The Curious Series Of Kempner And Irwin, arXiv:0806.4410 [math.CA], 2008-2015. [_Robert G. Wilson v_, Jun 01 2009]
Frank Irwin, A Curious Convergent Series, Amer. Math. Monthly, 23 (1916), 149-152.
A. D. Wadhwa, An interesting subseries of the harmonic series, Amer. Math. Monthly, 78 (1975), 931-933.
A. D. Wadhwa, Some convergent subseries of the harmonic series, Amer. Math. Monthly, 85 (1978), 661-663.
Eric Weisstein's World of Mathematics, Kempner Series.
Wikipedia, Kempner series
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series [_Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A052386, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838.

Cf. A052382 (zeroless numbers).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

More terms from Robert G. Wilson v, Jun 01 2009

A082831 Decimal expansion of Sum_{k >= 1, k has no digit 2 in base 10} 1/k.

Original entry on oeis.org

1, 9, 2, 5, 7, 3, 5, 6, 5, 3, 2, 8, 0, 8, 0, 7, 2, 2, 2, 4, 5, 3, 2, 7, 7, 6, 7, 7, 0, 1, 9, 4, 4, 5, 4, 1, 1, 5, 5, 2, 6, 0, 5, 3, 8, 3, 1, 1, 5, 4, 8, 7, 0, 1, 4, 9, 8, 6, 8, 3, 6, 2, 9, 4, 9, 1, 0, 4, 3, 0, 9, 0, 1, 6, 0, 1, 9, 5, 5, 1, 8, 0, 9, 2, 8, 0, 5, 4, 6, 2, 2, 1, 1, 2, 8, 4, 4, 2, 8, 6, 3, 5, 5, 6, 5

Offset: 2

Robert G. Wilson v, Apr 14 2003

Such sums are called Kempner series, see A082839 (analog for digit 0) for more information. - M. F. Hasler, Jan 13 2020

			19.25735653280807222453277677019445411552605383115487014986836294...

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Wikipedia, Kempner series.
Wolfram Library Archive, KempnerSums.nb (8.6 kB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series.

Cf. A002387, A024101, A052404 (numbers with no digit 2).

Cf. A082830, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 3, 4, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive *)

Equals Sum_{k in A052404\{0}} 1/k, where A052404 = numbers with no digit 2: these are omitted in the harmonic series. - M. F. Hasler, Jan 13 2020

More terms from Robert G. Wilson v, Jun 01 2009

A082832 Decimal expansion of Sum_{k >= 1, k has no digit 3 in base 10} 1/k.

Original entry on oeis.org

2, 0, 5, 6, 9, 8, 7, 7, 9, 5, 0, 9, 6, 1, 2, 3, 0, 3, 7, 1, 0, 7, 5, 2, 1, 7, 4, 1, 9, 0, 5, 3, 1, 1, 1, 4, 1, 4, 1, 5, 3, 8, 6, 9, 6, 7, 4, 7, 3, 0, 7, 8, 3, 4, 8, 9, 5, 0, 8, 5, 2, 8, 5, 0, 0, 2, 6, 7, 2, 9, 4, 9, 9, 6, 1, 9, 3, 8, 0, 3, 5, 0, 0, 5, 9, 0, 4, 7, 4, 9, 4, 0, 8, 0, 6, 0, 3, 5, 3, 4, 9, 8, 7, 9, 0

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 3 (A011533) have asymptotic density 1, i.e., almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - M. F. Hasler, Jan 13 2020

			20.569877950961230371075217419053111414153869674730783489508528500... - _Robert G. Wilson v_, Jun 01 2009

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. [From _Robert G. Wilson v_, Jun 01 2009]
Eric Weisstein's World of Mathematics, Kempner Series.
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series. [From _Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052405 (numbers with no '3'), A011533 (numbers with '3').

Cf. A082830, A082831, A082833, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 2, 4, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

Equals Sum_{k in A052405\{0}} 1/k, where A052405 = numbers with no digit 3. - M. F. Hasler, Jan 15 2020

More terms from Robert G. Wilson v, Jun 01 2009

A082833 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 4 in base 10} 1/k.

Original entry on oeis.org

2, 1, 3, 2, 7, 4, 6, 5, 7, 9, 9, 5, 9, 0, 0, 3, 6, 6, 8, 6, 6, 3, 9, 4, 0, 1, 4, 8, 6, 9, 3, 9, 5, 1, 2, 8, 4, 3, 7, 5, 0, 9, 5, 1, 7, 0, 3, 2, 7, 0, 0, 2, 1, 8, 1, 7, 2, 5, 1, 1, 8, 9, 5, 4, 1, 9, 7, 7, 8, 8, 4, 2, 7, 2, 4, 5, 1, 3, 3, 5, 3, 7, 5, 3, 8, 1, 2, 0, 1, 3, 0, 2, 8, 4, 0, 6, 9, 3, 5, 4, 7, 7, 8, 9, 7

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 4 (A011534) have asymptotic density 1, i.e., here almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - M. F. Hasler, Jan 13 2020

			21.32746579959003668663940148693951284375095170327002181725118954... - _Robert G. Wilson v_, Jun 01 2009

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. [From _Robert G. Wilson v_, Jun 01 2009]
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series [From _Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052406 (numbers with no 4), A011534 (numbers with a 4).

Cf. A082830, A082831, A082832, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 2, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive *) (* Robert G. Wilson v, Jun 01 2009 *)

sumpos(k=2,1/A052406(k)) \\ For illustration only, slow and not very precise: with \p19 takes 2 sec to get 5 digits right. - M. F. Hasler, Jan 13 2020

Equals Sum_{k in A052406\{0}} 1/k, where A052406 = numbers with no digit 3. - M. F. Hasler, Jan 15 2020

More terms from Robert G. Wilson v, Jun 01 2009

A082834 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 5 in base 10} 1/k.

Original entry on oeis.org

2, 1, 8, 3, 4, 6, 0, 0, 8, 1, 2, 2, 9, 6, 9, 1, 8, 1, 6, 3, 4, 0, 7, 2, 3, 5, 0, 4, 0, 6, 0, 9, 1, 8, 2, 7, 1, 7, 8, 4, 6, 5, 6, 7, 5, 1, 5, 0, 1, 3, 9, 1, 8, 2, 9, 1, 6, 7, 9, 3, 5, 9, 1, 8, 4, 2, 5, 0, 8, 6, 2, 6, 6, 8, 8, 2, 2, 9, 3, 8, 3, 5, 7, 7, 7, 2, 1, 3, 8, 3, 1, 9, 3, 2, 9, 2, 5, 4, 8, 8, 1, 3, 2, 4, 4

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 5 (A011535) have asymptotic density 1, i.e., here almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - M. F. Hasler, Jan 13 2020

			21.83460081229691816340723504060918271784656751501391829167935918... - _Robert G. Wilson v_, Jun 01 2009

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin arXiv:0806.4410 [math.CA], 2008-2015. [From _Robert G. Wilson v_, Jun 01 2009]
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series. [From _Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052413 (numbers with no '5'), A011535 (numbers with a '5').

Cf. A082830, A082831, A082832, A082833, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 2, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

Equals Sum_{k in A052413\{0}} 1/k, where A052413 = numbers with no digit 5. - M. F. Hasler, Jan 15 2020

More terms from Robert G. Wilson v, Jun 01 2009

Minor edits by M. F. Hasler, Jan 13 2020

A082835 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 6 in base 10} 1/k.

Original entry on oeis.org

2, 2, 2, 0, 5, 5, 9, 8, 1, 5, 9, 5, 5, 6, 0, 9, 1, 8, 8, 4, 1, 6, 7, 3, 8, 0, 4, 8, 0, 0, 0, 7, 5, 2, 7, 1, 0, 5, 1, 9, 3, 8, 5, 6, 1, 0, 6, 6, 6, 8, 4, 6, 3, 2, 7, 0, 2, 7, 6, 9, 3, 8, 2, 3, 3, 0, 5, 3, 2, 2, 8, 3, 5, 0, 8, 9, 1, 2, 4, 7, 5, 2, 6, 3, 4, 7, 7, 7, 6, 9, 9, 7, 4, 0, 5, 8, 9, 1, 4, 9, 3, 4, 4, 2, 5

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 6 (A011536) have asymptotic density 1, i.e., here almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - M. F. Hasler, Jan 13 2020

			22.20559815955609188416738048000752710519385610666846327027693823... - _Robert G. Wilson v_, Jun 01 2009

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2013. [From _Robert G. Wilson v_, Jun 01 2009]
Eric Weisstein's World of Mathematics, Kempner Series.
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series. [From _Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052414 (numbers with no '6'), A011536 (numbers with a '6').

Cf. A082830, A082831, A082832, A082833, A082834, A082836, A082837, A082838, A082839 (analog for digits 1, 2, 4, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

Equals Sum_{k in A052414\{0}} 1/k, where A052414 = numbers with no digit 6. - M. F. Hasler, Jan 15 2020

Minor edits by M. F. Hasler, Jan 13 2020

A082836 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 7 in base 10} 1/k.

Original entry on oeis.org

2, 2, 4, 9, 3, 4, 7, 5, 3, 1, 1, 7, 0, 5, 9, 4, 5, 3, 9, 8, 1, 7, 6, 2, 2, 6, 9, 1, 5, 3, 3, 9, 7, 7, 5, 9, 7, 4, 0, 0, 5, 9, 1, 5, 5, 4, 1, 6, 7, 2, 5, 1, 2, 3, 6, 1, 7, 9, 1, 4, 6, 0, 4, 4, 4, 0, 7, 1, 0, 5, 1, 2, 0, 0, 9, 5, 0, 7, 4, 0, 8, 5, 1, 4, 3, 2, 2, 2, 0, 8, 2, 3, 4, 5, 0, 0, 2, 1, 9, 1, 9, 2, 2, 5, 4

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 7 (A011537) have asymptotic density 1, i.e., here almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - M. F. Hasler, Jan 13 2020

			22.493475311705945398176226915339775974005915541672512361791460444... - _Robert G. Wilson v_, Jun 01 2009

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. [_Robert G. Wilson v_, Jun 01 2009]
Eric Weisstein's World of Mathematics, Kempner Series.
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series. [_Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052419 (numbers with no '7'), A011537 (numbers with a '7').

Cf. A082830, A082831, A082832, A082833, A082834, A082835, A082837, A082838, A082839 (analog for digits 1, 2, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

Equals Sum_{k in A052419\{0}} 1/k, where A052419 = numbers with no digit 7. - M. F. Hasler, Jan 14 2020

More terms from Robert G. Wilson v, Jun 01 2009

Minor edits by M. F. Hasler, Jan 13 2020

A082837 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 8 in base 10} 1/k.

Original entry on oeis.org

2, 2, 7, 2, 6, 3, 6, 5, 4, 0, 2, 6, 7, 9, 3, 7, 0, 6, 0, 2, 8, 3, 3, 6, 4, 4, 1, 5, 6, 7, 4, 2, 5, 5, 7, 8, 8, 9, 2, 1, 0, 7, 0, 2, 6, 1, 6, 3, 6, 0, 2, 1, 9, 8, 4, 3, 5, 3, 6, 3, 7, 6, 1, 6, 2, 4, 0, 0, 4, 6, 8, 2, 0, 1, 7, 5, 1, 3, 4, 8, 1, 2, 7, 0, 1, 0, 5, 6, 2, 1, 6, 5, 1, 5, 8, 9, 2, 2, 4, 7, 7, 5, 7, 9, 3

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 8 (A011538) have asymptotic density 1, i.e., here almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - M. F. Hasler, Jan 13 2020

			22.726365402679370602833644156742557889210702616360219843536376162... - _Robert G. Wilson v_, Jun 01 2009

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Eric Weisstein's World of Mathematics, Kempner Series.
Wikipedia, Kempner series.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series.

Cf. A002387, A024101, A052421 (numbers with no '8'), A011538 (numbers with a '8').

Cf. A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082838, A082839 (analog for digits 1, 2, 3, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

Equals Sum_{k in A052421\{0}} 1/k, where A052421 = numbers with no digit 8. - M. F. Hasler, Jan 14 2020

More terms and links from Robert G. Wilson v, Jun 01 2009

Minor edits by M. F. Hasler, Jan 13 2020

cp's OEIS Frontend

A007095 Numbers in base 9.

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A082830 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 1 in base 10} 1/k.

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A082839 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 0 in base 10} 1/k.

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A082831 Decimal expansion of Sum_{k >= 1, k has no digit 2 in base 10} 1/k.

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A082832 Decimal expansion of Sum_{k >= 1, k has no digit 3 in base 10} 1/k.

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A082833 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 4 in base 10} 1/k.

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