A082838 -id:A082838 - cp's OEIS frontend

A083896 Number of divisors of n with largest digit = 9 (base 10).

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, May 08 2003

			n = 117, 2 of the 6 divisors of 117 have largest digit = 9: {9,39}, therefore a(117) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A054055, A000005, A083888, A083889, A083890, A083891, A083892, A083893, A083894, A083895.

Cf. A001620, A082838.

f:= proc(n) nops(select(t -> max(convert(t, base, 10))=d, numtheory:-divisors(n))) end proc:
d:= 9:
map(f, [$1..200]); # Robert Israel, Oct 06 2019

Table[Count[Divisors[n],?(Max[IntegerDigits[#]]==9&)],{n,110}] (* _Harvey P. Dale, Mar 15 2018 *)

a(n) = A000005(n) - A083888(n) - A083889(n) - A083890(n) - A083891(n) - A083892(n) - A083893(n) - A083894(n) - A083895(n) = A000005(n) - A083903(n).

Sum_{k=1..n} a(k) ~ n * (log(n) + c), where c = 2*A001620 - 1 - A082838 = -22.766245... . - Amiram Eldar, Apr 17 2025

A140502 Decimal expansion of the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9.

Original entry on oeis.org

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

Offset: 2

Jonathan Vos Post, Jun 30 2008

In 1914, Kempner proved that the series 1/1 + 1/2 + ... + 1/8 + 1/10 + 1/11 + ... + 1/18 + 1/20 + 1/21 + ..., where the denominators are the positive integers that do not contain the digit 9, converges to a sum less than 90. (The actual sum is about 22.92068.) In 1916, Irwin proved that the sum of 1/n where n has at most a finite number of 9's is also a convergent series. We show how to compute sums of Irwin's series to high precision.

For example, the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ..., where the denominators have exactly one 9, is about 23.04428708074784831968. Note that this is larger than the sum of Kempner's "no 9" series. We also show how to construct nontrivial subseries of the harmonic series that have arbitrarily large, but computable, sums. For example, the sum of 1/n where n has at most 434 occurrences of the digit 0 is about 10016.32364577640186109739.

			23.04428708074784831968...

Robert Baillie, Summing the curious series of Kempner and Irwin

Cf. A082838.

(* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[9, 1, 111] (* Robert G. Wilson v, Aug 03 2010 *)

Offset corrected R. J. Mathar, Jan 26 2009

More terms from Robert G. Wilson v, Aug 03 2010

A179951 Decimal expansion of Sum_{k has exactly two bits equal to 1 in base 2} 1/k.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 03 2010

Obviously for k > 0 in base 2 having no bit equal to 1 the sum is 0 and for 1 bit equal to 1 the sum is 2.

			Sum_{k>0} 1/A018900(k) = 1.52899956069688841838263949451...

Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series

Cf. A065442, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839, A140502, A160502, A018900, A323482.

 evalf( 2*add( (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2), n = 1..18), 100); # Peter Bala, Jan 28 2022

(* first install irwinSums.m, see either reference, then *) First@ RealDigits@ iSum[1, 2, 2^7, 2]

Equals Sum_{j>=1} Sum_{i=0..j-1} 1/(2^i + 2^j).
From Amiram Eldar, Jun 30 2020: (Start)
Equals Sum_{k>=0} 1/(2^k + 1/2).
Equals 2 * A323482 - 1. (End)
Equals 2*Sum_{n >= 1} (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2). The first 18 terms of the series gives the constant correct to more than 100 decimal places. - Peter Bala, Jan 28 2022

A083903 Number of divisors of n with largest digit <= 8 (base 10).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 5, 2, 5, 1, 6, 4, 4, 2, 8, 3, 4, 3, 6, 1, 8, 2, 6, 4, 4, 4, 8, 2, 3, 3, 8, 2, 8, 2, 6, 5, 4, 2, 10, 2, 6, 4, 6, 2, 7, 4, 8, 3, 3, 1, 12, 2, 4, 5, 7, 4, 8, 2, 6, 3, 8, 2, 11, 2, 4, 6, 5, 4, 7, 1, 10, 4, 4, 2, 12, 4, 4, 3, 8, 1, 10, 3, 5, 3, 3, 2, 11, 1, 4, 4, 9, 2, 8

Offset: 1

Reinhard Zumkeller, May 08 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A054055, A000005, A007095, A082838, A083895, A083896, A083902.

Table[Length[DeleteCases[Divisors[n],?(Max[IntegerDigits[#]]==9&)]],{n,110}] (* _Harvey P. Dale, Nov 21 2015 *)

a(n) = A083902(n) + A083895(n) = A000005(n) - A083896(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A007095(k) = 22.920676... (A082838). - Amiram Eldar, Jan 04 2024

A375805 Decimal expansion of Sum_{n >= 1} 1/A171397(n).

Original entry on oeis.org

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

Offset: 2

Robert C. Lyons, Aug 29 2024

A variation on the harmonic series, in which the denominators are treated as base 11 numbers. Equivalently: sum of reciprocals of positive integers whose base-11 representation contains no digit A (no "10" digit).
Values were calculated using Mathematica code from Baillie & Schmelzer (see link). Note that the code in the Wolfram Library Archive, as it stands, does not support digits > 9 in bases > 10 (and doing the "obvious" thing will be interpreted as asking a different question with a different answer); the code was modified to support this.
Kempner (1914) showed that this series converges. - N. J. A. Sloane, Aug 31 2024
There is a slight ambiguity when we get to 1/10. This is to be regarded as 1/(1*11 + 0*1) = (1/11)-in-base-10 and not as 1/A =  1/(10*1) = (1/10)-in-base-10. - N. J. A. Sloane, Aug 30 2024

			26.2833282048814207699401516874442229241887980925...

Burnol, Jean-François. "Moments in the exact summation of the curious series of Kempner type." arXiv preprint arXiv:2402.08525 (2024).
A. J. Kempner, A Curious Convergent Series, American Mathematical Monthly, 21 (February, 1914), pp. 48-50. (https://dx.doi.org/10.2307/2972074)
Schmelzer, Thomas, and Robert Baillie. "Summing a curious, slowly convergent series." The American Mathematical Monthly 115.6 (2008): 525-540.

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 Baillie & Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Wolfram Library Archive
G. W. Brewster, An Old Result in a New Dress, The Mathematical Gazette, Vol. 37, No. 322 (Dec., 1953), pp. 269-270.
John D. Cook, A strange take on the harmonic series, Blog, 29 August 2024.
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]
Eric Weisstein's World of Mathematics, Kempner Series.
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]

Cf. A171397, A375523/A375524.

Corrected data provided by Gareth McCaughan, Sep 02 2024

A111935 Numerator of n-th term of the harmonic series after removal of all terms 1/m from Sum_{m=1..n} 1/m for which m contains a 9 in its decimal representation.

Original entry on oeis.org

1, 3, 11, 25, 137, 49, 363, 761, 789, 8959, 27647, 368651, 377231, 128413, 261831, 4531207, 41461543, 8414831, 8531519, 8642903, 201237217, 203585563, 5145999379, 5200191979, 15757132337, 15908097437, 16048998197, 501745966907

Offset: 1

Reinhard Zumkeller, Aug 22 2005

Denominator = A111936;
Lim_{n->infinity} a(n)/A111936(n) = C < 80.
The sum of the harmonic series after removing all terms containing a 9 in decimal representation in decimal system converges and the sum is < 80. Hence the sum of the harmonic series in which at least one digit is missing (from 0 to 9) converges and the sum is less than 810.

			n=9: 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/10 = 789/280, therefore a(9) = 789.

G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 3, sect. 4, Problem 124.
Jason Earls and Amarnath Murthy, Some fascinating variations in harmonic series, Octogon Mathematical Magazine, Vol. 12, No. 2, 2004.

Cf. A001008, A007095, A082838, A111936 (denominators).

a:=[k:k in [1..100]| not 9 in Intseq(k)]; [Numerator( &+[1/a[m]: m in [1..n]]): n in [1..30] ];  // Marius A. Burtea, Dec 30 2019

Definition edited by N. J. A. Sloane, Dec 30 2019

A111936 Denominator of n-th term of the harmonic series after removal of all terms 1/m from Sum_{m=1..n} 1/m for which m contains a 9 in its decimal representation.

Original entry on oeis.org

1, 2, 6, 12, 60, 20, 140, 280, 280, 3080, 9240, 120120, 120120, 40040, 80080, 1361360, 12252240, 2450448, 2450448, 2450448, 56360304, 56360304, 1409007600, 1409007600, 4227022800, 4227022800, 4227022800, 131037706800, 262075413600

Offset: 1

Reinhard Zumkeller, Aug 22 2005

Numerator = A111935;

lim_{n->infinity} A111935(n)/a(n) = C < 80.

			n=9: 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/10 = 789/280, therefore a(9) = 280.

G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 3, sect. 4, Problem 124.

Cf. A002805, A007095, A082838, A111935 (numerators).

a:=[k:k in [1..100]| not 9 in Intseq(k)]; [Denominator( &+[1/a[m]: m in [1..n]]): n in [1..30] ]; // Marius A. Burtea, Dec 29 2019

Denominator[Accumulate[DeleteCases[Table[1/n,{n,40}],?(MemberQ[ IntegerDigits[ Denominator[#]],9]&)]]] (* _Harvey P. Dale, Mar 05 2013 *)

Definition edited by N. J. A. Sloane, Dec 30 2019

A338287 Decimal expansion of the sum of reciprocals of the numbers that are not pandigital numbers (version 2, A171102).

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Oct 20 2020

The sum of the reciprocals of the terms of the complement of A171102: numbers with at most 9 distinct digits. It is the union of the 10 sequences of numbers without a single given digit (see the Crossrefs section).

The terms in the data section were taken from the 200 decimal digits given by Strich and Müller (2020).

			65.74331110185328196734583167680868411685344106635398...

Robert Strich and Eric Müller, Ziffernreduzierte Zahlen, in: E. Specht, E. Quaisser and P. Bauermann (eds.), 50 Jahre Bundeswettbewerb Mathematik, Springer Spektrum, Berlin, Heidelberg, 2020, pp. 171-182.

Cf. A171102, A179954, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839.

Cf. A052382 (numbers without the digit 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), A007095 (without 9).

Equals 1/1 + 1/2 + 1/3 + ... + 1/1023456788 + 1/1023456790 + ..., i.e., A171102(1) = 1023456789 is the first number whose reciprocal is not in the sum.

cp's OEIS Frontend

A083896 Number of divisors of n with largest digit = 9 (base 10).

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A140502 Decimal expansion of the sum of the series 1/9 + 1/19 + 1/29 + 1/39 + 1/49 + ... where the denominators have exactly one 9.

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A179951 Decimal expansion of Sum_{k has exactly two bits equal to 1 in base 2} 1/k.

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A083903 Number of divisors of n with largest digit <= 8 (base 10).

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A375805 Decimal expansion of Sum_{n >= 1} 1/A171397(n).

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A111935 Numerator of n-th term of the harmonic series after removal of all terms 1/m from Sum_{m=1..n} 1/m for which m contains a 9 in its decimal representation.

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A111936 Denominator of n-th term of the harmonic series after removal of all terms 1/m from Sum_{m=1..n} 1/m for which m contains a 9 in its decimal representation.

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A338287 Decimal expansion of the sum of reciprocals of the numbers that are not pandigital numbers (version 2, A171102).

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