A024101 -id:A024101 - cp's OEIS frontend

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

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

A366661 Number of divisors of 9^n-1.

Original entry on oeis.org

4, 10, 16, 24, 24, 80, 16, 112, 128, 180, 64, 384, 16, 160, 768, 256, 128, 1280, 64, 864, 768, 640, 32, 14336, 384, 160, 4096, 1536, 256, 23040, 128, 576, 2048, 1280, 768, 12288, 128, 640, 12288, 16128, 128, 61440, 32, 12288, 196608, 320, 512, 131072, 2048

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

			a(2)=10 because 9^2-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

Max Alekseyev, Table of n, a(n) for n = 1..690

Cf. A024101, A000005, A057952, A366660, A366662, A366663.

Cf. A046801, A366575, A366602, A366612, A366621, A366633, A366652, A070528, A366683, A366709.

a:=n->numtheory[tau](9^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 9^Range[100]-1]
```
PARI
```
a(n) = numdiv(9^n-1);
```

a(n) = sigma0(9^n-1) = A000005(A024101(n)).

a(n) = A366575(2*n) = A366575(n) * A366577(n) * (4 + A007814(n)) / (2 * (3 + A007814(n))). - Max Alekseyev, Jan 07 2024

A366662 Sum of the divisors of 9^n-1.

Original entry on oeis.org

15, 186, 1680, 15876, 123690, 1541568, 8992680, 111757968, 967814400, 9366647892, 62424587520, 852903426816, 4766016364260, 55176998178240, 550081165885440, 4829754617483040, 31725040326819840, 471309320999516160, 2535353780263288800, 33995669076586206864

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

			a(2)=186 because 9^2-1 has divisors {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}.

Max Alekseyev, Table of n, a(n) for n = 1..690

Cf. A024101, A000203, A366660, A366661, A366663.

Cf. A075708, A366576, A366603, A366613, A366622, A366634, A366653, A102146, A366684, A366710.

a:=n->numtheory[sigma](9^n-1):
seq(a(n), n=1..30);

Mathematica
```
DivisorSigma[1, 9^Range[30]-1]
```

a(n) = sigma(9^n-1) = A000203(A024101(n)).

a(n) = A366576(2*n) = A366576(n) * A366578(n) * (2^(4 + A007814(n)) - 1) / (2^(3 + A007814(n)) - 1) / 3. - Max Alekseyev, Jan 07 2024

A052379 Number of integers from 1 to 10^(n+1)-1 that lack 0 and 1 as a digit.

Original entry on oeis.org

8, 72, 584, 4680, 37448, 299592, 2396744, 19173960, 153391688, 1227133512, 9817068104, 78536544840, 628292358728, 5026338869832, 40210710958664, 321685687669320, 2573485501354568, 20587884010836552, 164703072086692424, 1317624576693539400, 10540996613548315208

Offset: 0

Odimar Fabeny, Mar 12 2000

			For n=1, the numbers from 1 to 99 which have 0 or 1 as a digit are the numbers 1 and 10, 20, 30, ..., 90 and 11, 12, ..., 18, 19 and 21, 31, ..., 91. So a(1) = 99 - 27 = 72.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A023001, A024101, A052386.

[(8^(n+2)-1)/7-1: n in [0..20]]; // Vincenzo Librandi, Jul 04 2011

A052379:=n->(8^(n+2)-1)/7-1: seq(A052379(n), n=0..20); # Wesley Ivan Hurt, Sep 26 2014

(8^(Range[0,20]+2)-1)/7-1 (* or *) LinearRecurrence[{9,-8},{8,72},20] (* Harvey P. Dale, Sep 22 2013 *)

a(n)=8^(n+2)\7 - 1 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = (8^(n+2) - 1)/7 - 1.
G.f.: 8/((1-x)*(1-8*x)). - R. J. Mathar, Nov 19 2007
a(n) = 8*a(n-1) + 8.
a(n) = Sum_{k=1..n} 8^k. - corrected by Michel Marcus, Sep 25 2014
Conjecture: a(n) = A023001(n+2)-1. - R. J. Mathar, May 18 2007. Comment from Vim Wenders, Mar 26 2008: The conjecture is true: the g.f. leads to the closed form a(n) = -(8/7)*(1^n) + (64/7)*(8^n) = (-8 + 64*8^n)/7 = (-8 + 8^(n+2))/7 = (8^(n+2) - 1)/7 - 1 = A023001(n+2) - 1.
a(n) = 9*a(n-1) - 8*a(n-2); a(0)=8, a(1)=72. - Harvey P. Dale, Sep 22 2013
a(n) = 8*A023001(n+1). - Alois P. Heinz, Feb 15 2023

More terms and revised description from James Sellers, Mar 13 2000

cp's OEIS Frontend

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

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

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

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