A002387 -id:A002387 - cp's OEIS frontend

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

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

A056053 a(n) = smallest odd number 2m+1 such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

Original entry on oeis.org

1, 3, 15, 113, 837, 6183, 45691, 337607, 2494595, 18432707, 136200301, 1006391657, 7436284415, 54947122715, 406007372211, 3000011249847, 22167251422541, 163795064320249, 1210290918990281, 8942907496445513, 66079645178783351, 488266205223462461, 3607826381608149807

Offset: 0

Robert G. Wilson v, Jul 25 2000 and Jan 11 2004

nonn

a(2) = 15 and a(3) = 113 are related to the Borwein integrals. Concretely, a(2) = 15 is the smallest odd m such that the integral Integral_{x=-oo..oo} Product_{1<=k<=m, k odd} (sin(k*x)/(k*x)) dx is slightly less than Pi, and a(3) = 113 is the smallest odd m such that the integral Integral_{x=-oo..oo} cos(x) * Product_{1<=k<=m, k odd} (sin(k*x)/(k*x)) dx is slightly less than Pi/2. See the Wikipedia link and the 3Blue1Brown video link below. - Jianing Song, Dec 10 2022

Calvin C. Clawson, "Mathematical Mysteries, The Beauty and Magic of Numbers," Plenum Press, NY and London, 1996, page 64.

Robert G. Wilson v, Table of n, a(n) for n = 0..500
Grant Sanderson, Researchers thought this was a bug (Borwein integrals), 3Blue1Brown video (2022).
Wikipedia, Borwein integral

Cf. A002387, A056054, A091463, A091464, A091465.

Cf. also A092318, A092317, A092315, A281355, A074599, A025547.

s = 0; k = 1; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 11}]

a(n) ~ floor((1/2)*A002387(2n)).
The next term is approximately the previous term * e^2.
a(n) = A092315(n)*2 + 1 = floor(exp(n*2-Euler)/4+1/8)*2+1 for all n (conjectured). - M. F. Hasler, Jan 24 2017
a(n) ~ exp(2*n - A350763) = (1/2)*exp(2*n - gamma), gamma = A001620. - A.H.M. Smeets, Apr 15 2022

Corrected by N. J. A. Sloane, Feb 16 2004
More terms from Robert G. Wilson v, Apr 17 2004
a(17) corrected - see correction in A092315. - Gerhard Kirchner, Jul 25 2020
a(0) prepended by Robert G. Wilson v, Oct 23 2024

A055980 a(n) = floor(Sum_{i=1..n} 1/i).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Henry Bottomley, Jul 20 2000

nonn

If we choose at random (uniformly) a permutation in the symmetric group S_n then a(n) is the expected number of cycles (rounded down) in the cycle decomposition of the permutation. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 17 2001

a(n) = A214075(n,n-1) for n > 0. - Reinhard Zumkeller, Jul 03 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
L. D. Kudryavtsev, Harmonic series, The Encyclopedia of Mathematics.
Eric Weisstein's World of Mathematics, High-Water Mark

Cf. A002387, A004080 (indices of records).

import Data.Ratio ((%), denominator)
a055980 = floor . sum . map (1 %) . enumFromTo 1
a055980_list = map floor $ scanl1 (+) $ map (1 %) [1..]
-- Reinhard Zumkeller, Jul 03 2012

Floor[HarmonicNumber[Range[110]]] (* Harvey P. Dale, May 22 2021 *)

a(n) ~ log(n) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001

a(n) = floor[A001008(n)/A002805(n)]. - Lekraj Beedassy, Sep 17 2006

A092315 a(n) is the smallest m such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

Original entry on oeis.org

1, 7, 56, 418, 3091, 22845, 168803, 1247297, 9216353, 68100150, 503195828, 3718142207, 27473561357, 203003686105, 1500005624923, 11083625711270, 81897532160124, 605145459495140, 4471453748222756, 33039822589391675, 244133102611731230, 1803913190804074903

Offset: 1

N. J. A. Sloane, Feb 16 2004

nonn

From Gerhard Kirchner, May 21 2020: (Start)
The terms a(n), evaluated by the formula, should pass the test OH(a(n))=n and OH(a(n)-1)=n-1, where OH(m) is the odd harmonic series, see above.
Another formula, see link Asymptotic formulas, formula 1, is OH(m) = (log(4*m)+gamma)/2+1/(2*m)-11/(48*m^2)+1/(8*m^3)-127*t/(1920*m^4), 0The Maxima code includes both tests and creates a b-file in the current directory. For n<=1000, the case "Precision too low" does not occur. (End)
a(2) = 7 and a(3) = 56 are related to the Borwein integrals. Concretely, a(2) = 7 is the smallest m such that the integral Integral_{x=-oo..oo} Product_{k=0..m} (sin((2*k+1)*x)/((2*k+1)*x)) dx is slightly less than Pi, and a(3) = 56 is the smallest m such that the integral Integral_{x=-oo..oo} cos(x) * Product_{k=0..m} (sin((2*k+1)*x)/((2*k+1)*x)) dx is slightly less than Pi/2. See the Wikipedia link and the 3Blue1Brown video link below. - Jianing Song, Dec 10 2022

Gerhard Kirchner, Table of n, a(n) for n = 1..1000
Gerhard Kirchner, Asymptotic formulas
Grant Sanderson, Researchers thought this was a bug (Borwein integrals), 3Blue1Brown video (2022).
Wikipedia, Borwein integral

Except for first term, same as A092318. Equals (A056053-1)/2.

Cf. A074599, A025547, A281355.

A092315[n_] := Floor[Exp[2*n - EulerGamma]/4]; Table[A092315[n], {n, 1, 22}] (* Robert P. P. McKone, Jul 13 2021 *)

block(
fpprec:1000, gam: %gamma, nmax:1000,
fl: openw("bfile1000.txt"),
OH(k,t):=(log(4*k)+gam)/2+1/(2*k)-11/(48*k^2)+1/(8*k^3)-127*t/(1920*k^4),
printf(fl, "1 1"),   newline(fl),
for n from 2 thru nmax do
(u: bfloat(exp(2*n-gam)/4), k: floor(u),
x0: bfloat(OH(k,0)), x01: bfloat(OH(k,1)), x1: bfloat(OH(k-1,0)),
n0: floor(x0), n01: floor(x01), n1: floor(x1),  m: n,
if n0=n and n01=n and n1=n-1 then
         (h: concat(n, " ", k), printf(fl, h),  newline(fl)) else n: nmax),
if mGerhard Kirchner, Jul 23 2020 */
/* The first nmax terms are saved as a b-file */

a(n) = floor(exp(2*n-gamma)/4+1/8) for all n >= 1 (conjectured; see also comments in A002387). - M. F. Hasler, Jan 22 2017

a(n) = floor(exp(2*n-gamma)/4). - Gerhard Kirchner, Jul 23 2020

More terms from M. F. Hasler, Jan 24 2017

a(17) in the data section and 127 terms in the b-file corrected by Gerhard Kirchner, Jul 23 2020

A074631 a(n) is the smallest k such that the sum of the first k terms of the composite-harmonic series, Sum_{j=1..k} 1/(j-th composite), is > n.

Original entry on oeis.org

9, 44, 168, 587, 1940, 6192, 19285, 59010, 178122, 531923, 1574706, 4628338, 13521477, 39299115, 113712434, 327752962, 941457955, 2696114317, 7700146599, 21938239766

Offset: 1

Labos Elemer, Aug 27 2002

			1/4 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/14 + 1/15 + 1/16 = 1045/1008, but if 1/16 is not present, the sum is less than 1; 16 is the ninth composite number, so a(1) = 9.

Cf. A002387, A002808, A004080, A016088, A046024, A065855, A076751.

NextComposite[n_] := Block[{k = n + 1}, While[PrimeQ[k], k++ ]; k]; s=0; k = 4; Do[While[s = s + 1/k; s < n, k = NextComposite[k]]; Print[k - PrimePi[k] - 1]; k = NextComposite[k], {n, 1, 20}]
Table[Position[Accumulate[1/Select[Range[5*10^6],CompositeQ]],?(#>n&),1,1],{n,12}]//Flatten (* The program generates the first 12 terms of the sequence. *) (* _Harvey P. Dale, Jan 22 2023 *)

lista(cmax) = {my(n = 1, s = 0, k = 0); forcomposite(c = 1, cmax, k++; s += 1/c; if(s > n, print1(k, ", "); n++));} \\ Amiram Eldar, Jul 17 2024

a(n) = Min { k : Sum_{j=1..k} 1/A002808(j) > n }.
Limit_{n->oo} a(n+1)/a(n) = e. - Robert G. Wilson v, Aug 28 2002
a(n) = A065855(A076751(n)). - Amiram Eldar, Jul 17 2024

Edited by Robert G. Wilson v, Aug 28 2002
More terms from Robert Gerbicz, Aug 30 2002
2 more terms from Robert G. Wilson v, Sep 03 2002
Edited by Jon E. Schoenfield, Sep 13 2023
a(18)-a(20) from Amiram Eldar, Jul 17 2024

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

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