A002387 -id:A002387 - cp's OEIS frontend

A242654 1 followed by the union of the terms > 2 in A002387 (or A004080) and A115515.

1, 3, 4, 10, 11, 30, 31, 82, 83, 226, 227, 615, 616, 1673, 1674, 4549, 4550, 12366, 12367, 33616, 33617, 91379, 91380, 248396, 248397, 675213, 675214, 1835420, 1835421, 4989190, 4989191, 13562026, 13562027, 36865411, 36865412, 100210580, 100210581, 272400599, 272400600, 740461600, 740461601, 2012783314

Offset: 1

N. J. A. Sloane, May 29 2014

nonn

Ray Chandler, May 29 2014, proposes this as the most likely continuation of A079353.

Cf. A002387, A004080, A115515, A079353.

b[n_] := Ceiling[k /. FindRoot[HarmonicNumber[k] == n, {k, Exp[n]}, WorkingPrecision -> 100]] - 1;
bb = Array[b, 22];
A242654 = Union[bb, bb + 1] // Rest (* Jean-François Alcover, Apr 10 2019 *)

A374663 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1 / (k*a(k)) < 1.

Original entry on oeis.org

2, 2, 2, 4, 10, 201, 34458, 1212060151, 1305857607493406801, 1534737681943564047120326770001682121, 2141290683979549415450148346297540185977813099483710032048213090481251382

Offset: 1

Rémy Sigrist, Aug 04 2024

nonn

The harmonic series, Sum_{k > 0} 1/k, diverges. We divide each of its terms in such a way as to have a series bounded by 1.

			The initial terms, alongside the corresponding sums, are:
  n  a(n)        Sum_{k=1..n} 1/(k*a(k))
  -  ----------  -----------------------------------------
  1           2  1/2
  2           2  3/4
  3           2  11/12
  4           4  47/48
  5          10  1199/1200
  6         201  241199/241200
  7       34458  9696481199/9696481200
  8  1212060151  11752718467440661199/11752718467440661200
...
The denominators are in A375516.

Rémy Sigrist and N. J. A. Sloane, Dampening Down a Divergent Series, Manuscript in preparation, September 2024.

N. J. A. Sloane, Table of n, a(n) for n = 1..14
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A000058, A001008/A002805, A002387, A374983, A375516, A375517.

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(n*a(n))) end:
a:= proc(n) a(n):= 1+floor(1/((1-s(n-1))*n)) end:
seq(a(n), n=1..11);  # Alois P. Heinz, Oct 18 2024

s[n_] := s[n] = If[n == 0, 0, s[n - 1] + 1/(n*a[n])];
a[n_] := 1 + Floor[1/((1 - s[n - 1])*n)];
Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

{ t = 0; for (n = 1, 11, for (v = ceil(1/(n*(1-t))), oo, if (t + 1/(n*v) < 1, t += 1/(n*v); print1 (v", "); break;););); }

from itertools import count, islice
from math import gcd
def A374663_gen(): # generator of terms
    p, q = 0, 1
    for k in count(1):
        yield (m:=q//(k*(q-p))+1)
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
A374663_list = list(islice(A374663_gen(),11)) # Chai Wah Wu, Aug 28 2024

The ratios a(n)^2/a(n+1) are very close to the values 2, 2, 1, 8/5, 1/2, 7/6, 48/49, 9/8, 10/9, 11/10, 24/11^2, 13/12, 56/13^2, ... So it seems that often (but not always), a(n+1) is very close to (n/(n+1))*a(n)^2. - N. J. A. Sloane, Sep 08 2024

A004080 Least k such that H(k) >= n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

Original entry on oeis.org

0, 1, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, 675214, 1835421, 4989191, 13562027, 36865412, 100210581, 272400600, 740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245429, 298723530401, 812014744422

Offset: 0

N. J. A. Sloane, Clark Kimberling

			a(2)=4 because 1/1 + 1/2 + 1/3 + 1/4 > 2.

Bruno Rizzi and Cristina Scagliarini: I numeri armonici. Periodico di matematiche, "Mathesis", pp. 17-58, 1986, numbers 1-2. [From Vincenzo Librandi, Jan 05 2009]
W. Sierpiński, Sur les décompositions de nombres rationnels, Oeuvres Choisies, Académie Polonaise des Sciences, Warsaw, Poland, 1974, p. 181.
N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.

T. D. Noe, Table of n, a(n) for n=0..100 (using Hickerson's formula in A002387)
John V. Baxley, Euler's constant, Taylor's formula, and slowly converging series, Math. Mag. 65 (1992), 302-313.
R. P. Boas, Jr. and J. W. Wrench, Jr., Partial sums of the harmonic series, Amer. Math. Monthly, 78 (1971), 864-870.
Keneth Adrian Dagal, A Lower Bound for tau(n) for k-Multiperfect Number, arXiv:1309.3527 [math.NT], 2013.
J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number
Eric Weisstein's World of Mathematics, Harmonic Series
Eric Weisstein's World of Mathematics, High-Water Mark

Apart from first two terms, same as A002387.

import Data.List (findIndex); import Data.Maybe (fromJust)
a004080 n = fromJust $
   findIndex (fromIntegral n <=) $ scanl (+) 0 $ map recip [1..]
-- Reinhard Zumkeller, Jul 13 2014

aux[0] = 0; Do[aux[n] = Floor[Floor[Sum[1/i, {i, n}]]]; If[aux[n] > aux[n - 1], Print[n]], {n, 1, 14000}] (* José María Grau Ribas, Feb 20 2010 *)
a[0] = 0; a[1] = 1; a[n_] := k /. FindRoot[ HarmonicNumber[k] == n, {k, Exp[n - EulerGamma]}, WorkingPrecision -> 50] // Ceiling; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Aug 13 2013, after Charles R Greathouse IV *)

my(t=0, n=0); for(i=0, 10^20, if (i, t+=1./i); if(t>=n, print1(i, ", "); n++)) \\ Thomas Gettys (tpgettys(AT)comcast.net), Jan 21 2007; corrected by Michel Marcus, Jan 19 2022

Limit_{n->oo} a(n+1)/a(n) = exp(1). - Sébastien Dumortier, Jun 29 2005
a(n) = exp(n - gamma + o(1)). - Charles R Greathouse IV, Mar 10 2009
a(n) = A002387(n) for n>1. - Robert G. Wilson v, Jun 18 2015

Terms for n >= 13 computed by Eric W. Weisstein; corrected by James R. Buddenhagen and Eric W. Weisstein, Feb 18 2001
Edited by Dean Hickerson, Apr 19 2003
More terms from Sébastien Dumortier, Jun 29 2005
a(27) from Thomas Gettys (tpgettys(AT)comcast.net), Dec 05 2006
a(28) from Thomas Gettys (tpgettys(AT)comcast.net), Jan 21 2007
Edited by Charles R Greathouse IV, Mar 24 2010

A082838 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 9 in base 10} 1/k.

Original entry on oeis.org

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

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 9 (A011539) 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

			22.920676619264150348163657094375931914944762436998481568541998356... - _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.
Aubrey J. Kempner, A Curious Convergent Series, American Mathematical Monthly, volume 21, number 2, February 1914, pages 48-50. Or JSTOR.
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, A007095 (numbers with no '9'), A011539 (numbers with a '9'), A024101.

Cf. A082830 .. A082839 (analog for digits 1, ..., 8 and 0), A140502.

(* see the Mmca in Wolfram Library Archive link *)

Equals Sum_{k in A007095\{0}} 1/k, where A007095 = numbers with no digit 9. - M. F. Hasler, Jan 15 2020

More terms from Robert G. Wilson v, Apr 14 2009
More terms from Robert G. Wilson v, Jun 01 2009
Minor edits 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

A227631 Array t(n,k): row n consists of the positive integers m for which the least splitter of H(m) and H(m+1) is n, where H denotes harmonic number.

Original entry on oeis.org

1, 3, 2, 10, 6, 5, 30, 18, 7, 4, 82, 50, 15, 8, 16, 226, 136, 21, 13, 20, 9, 615, 372, 42, 23, 24, 12, 14, 1673, 1014, 59, 38, 36, 25, 19, 44, 4549, 2758, 115, 64, 45, 35, 22, 56, 17, 12366, 7500, 161, 106, 55, 70, 26, 73, 33, 11, 33616, 20389, 315, 175, 67

Offset: 1

Clark Kimberling, Jul 18 2013

Suppose that x < y. The least splitter of x and y is introduced here as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Conjecture: every row of the array in A227631 is infinite, and every positive integer occurs exactly once. Let r be the limiting ratio of consecutive terms of row 1; is r = e?

For any reduced fraction c/d with c sufficiently large, there exists an (H(m),H(m+1)) pair bracketing it to within 1/LCM(1,2,3,...,d), so d is the least splitter for all such pairs, and every row is infinite. Since each (H(m),H(m+1)) pair is assigned to a single row, each positive integer m occurs exactly once by construction. Since t(1,k) = A002387(k) - 1 for all k >= 1, r = e is indeed the limiting ratio for row 1. - Matthew House, Aug 14 2024

			Northwest corner of the array:
  1 ... 3 ... 10 ... 30 ... 82 ... 226
  2 ... 6 ... 18 ... 50 ... 136 .. 372
  5 ... 7 ... 15 ... 21 ... 42 ... 59
  4 ... 8 ... 13 ... 23 ... 38 ... 64
  16 .. 20 .. 24 ... 36 ... 45 ... 55
  9 ... 12 .. 25 ... 35 ... 70 ... 97
  14 .. 19 .. 22 ... 26 ... 34 ... 40
t(2,1) = 2 matches 1 + 1/2 <= 3/2 < 1 + 1/2 + 1/3;
similarly, t(2,2) = 6 matches H(6) <= 5/2 < H(7) and t(2,3) = 18 matches H(18) <= 7/2 < H(19).

Matthew House, Table of n, a(n) for n = 1..10011 (rows 1..141).

Cf. A227629, A227630.

h[n_] := h[n] = HarmonicNumber[n]; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; t = Table[r[h[n], h[n + 1]], {n, 1, 40000}];
d = Denominator[t]; u[n_] := Flatten[Position[d, n]]; TableForm[Table[u[n], {n, 1, 50}]]  (* A227631  *)
r1[n_, k_] := u[n][[k]]; z = 11;  v = Flatten[Table[r1[n - k + 1, k], {n, z}, {k, n, 1, -1}]]  (*  A227631 sequence *)  (* Peter J. C. Moses, Jul 15 2013 *)

A019529 Sum of a(n) terms of 1/sqrt(k) first strictly exceeds n.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60, 68, 76, 85, 95, 105, 115, 126, 138, 150, 162, 175, 189, 202, 217, 232, 247, 263, 280, 297, 314, 332, 351, 370, 389, 409, 430, 451, 472, 494, 517, 540, 563, 587, 612, 637, 662, 688, 715, 741, 769, 797, 825

Offset: 0

Robert G. Wilson v

nonn

			Let b(k) = 1 + 1/sqrt(2) + 1/sqrt(3) + ... + 1/sqrt(k):
.k.......1....2.....3.....4.....5.....6.....7
-------------------------------------------------
b(k)...1.00..1.71..2.28..2.78..3.23..3.64..4.01
For A019529 we have:
  n=0: smallest k is a(0) = 1 since 1.00 > 0
  n=1: smallest k is a(1) = 2 since 1.71 > 1
  n=2: smallest k is a(2) = 3 since 2.28 > 2
  n=3: smallest k is a(3) = 5 since 3.23 > 3
  n=4: smallest k is a(4) = 7 since 4.01 > 4
For A054040 we have:
  n=1: smallest k is a(1) = 1 since 1.00 >= 1
  n=2: smallest k is a(2) = 3 since 2.28 >= 2
  n=3: smallest k is a(3) = 5 since 3.23 >= 3
  n=4: smallest k is a(4) = 7 since 4.01 >= 4

A054040 is another version. See also A002387, A004080.

s = 0; k = 1; Do[ While[ s <= n, s = s + N[ 1/Sqrt[ k ], 24 ]; k++ ]; Print[ k - 1 ], {n, 1, 75} ]
With[{c=N[Accumulate[Table[1/Sqrt[x],{x,1000}]]]},Table[Position[c,?(#>n&),1,1],{n,0,1000}]]//Flatten (* _Harvey P. Dale, Mar 19 2025 *)

Edited by N. J. A. Sloane, Sep 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

A056903 Numbers n such that the numerator of the rational number 1 + 1/2 + 1/3 + ... + 1/n is a prime number.

Original entry on oeis.org

2, 3, 5, 8, 9, 21, 26, 41, 56, 62, 69, 79, 89, 91, 122, 127, 143, 167, 201, 230, 247, 252, 290, 349, 376, 459, 489, 492, 516, 662, 687, 714, 771, 932, 944, 1061, 1281, 1352, 1489, 1730, 1969, 2012, 2116, 2457, 2663, 2955, 3083, 3130, 3204, 3359, 3494, 3572

Offset: 1

James R. Buddenhagen, Feb 23 2001

nonn

Related to partial sums of the harmonic series and to Wolstenholme's Theorem.

Some of the larger entries may only correspond to probable primes.

			5 is in this sequence because 1+1/2+1/3+1/4+1/5 = 137/60 and 137 is prime.

Eric Weisstein, Table of n, a(n) for n = 1..97
J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A002387, A004080.

Cf. A001008 (numerator of the harmonic number H(n)), A067657 (primes that are the numerator of a harmonic number).

Select[Range[1000], PrimeQ[Numerator[HarmonicNumber[ # ]]] &]

isok(n) = isprime(numerator(sum(k=1, n, 1/k))); \\ Michel Marcus, Feb 05 2016

use ntheory ":all"; for (1..1000) { say if is_prime((harmfrac($))[0]); } # _Dana Jacobsen, Feb 05 2016

Terms from 201 to 492 computed by Jud McCranie.
More terms from Kamil Duszenko (kdusz(AT)wp.pl), Jun 22 2003
29 more terms from T. D. Noe, Sep 15 2004
Further terms found by Eric W. Weisstein, Mar 07 2005, Mar 29 2005, Nov 28 2005, Sep 23 2006

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

cp's OEIS Frontend

A242654 1 followed by the union of the terms > 2 in A002387 (or A004080) and A115515.

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A374663 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1 / (k*a(k)) < 1.

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A004080 Least k such that H(k) >= n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

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A082838 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 9 in base 10} 1/k.

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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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A227631 Array t(n,k): row n consists of the positive integers m for which the least splitter of H(m) and H(m+1) is n, where H denotes harmonic number.

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A019529 Sum of a(n) terms of 1/sqrt(k) first strictly exceeds n.

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