A082830 -id:A082830 - cp's OEIS frontend

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

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

A194181 Decimal expansion of the (finite) value of Sum_{k >= 1, k has no even digit in base 10 } 1/k.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 18 2011

For an elementary proof that this series is convergent, see Honsberger's reference. - Bernard Schott, Jan 13 2022

			3.17176547341590495722870970875061165679705070839628572416418689843...

Ross Honsberger, Mathematical Gems II, Dolciani Mathematical Expositions No. 2, Mathematical Association of America, 1976, pp. 102 and 177.

Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Thomas Schmelzer and Robert Baillie, Summing a curious, slowly convergent, harmonic subseries, American Mathematical Monthly 115:6 (2008), pp. 525-540; preprint.
Wikipedia, Kempner series.

Cf. A014261, A082830, A194182.

RealDigits[kSum[{0, 2, 4, 6, 8}, 120 ]][[1]] (* Amiram Eldar, Jun 15 2023, using Baillie and Schmelzer's kempnerSums.nb, see Links *)

Equals Sum_{n>=1} 1/A014261(n). - Bernard Schott, Jan 13 2022

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

A375533 a(n) = numerator of Sum_{i=1..n} 1/A038603(i).

Original entry on oeis.org

0, 1, 5, 31, 247, 5891, 175669, 6639823, 290694979, 13885515383, 746406329689, 44593096214321, 3020489689357037, 222690147603898211, 17752712881208877899, 1486130275559909484787, 133315968357656471537153, 13025132201814060676912631, 2913672358303309675918969343, 663425761972477930761347977351, 152383524508438692136746106396609

Offset: 0

N. J. A. Sloane, Sep 06 2024

Numerator of sum of reciprocals of primes with no digit "1".

Of interest because it appears that the value of Sum_{i=1..oo} 1/A038603(i) = lim_{i->oo} a(i)/A375534(i) is extremely difficult to compute - so difficult that its decimal expansion does not have an OEIS entry. (Compare A082830.)

			The first few sums are 0/1, 1/2, 5/6, 31/30, 247/210, 5891/4830, 175669/140070, 6639823/5182590,  ...

Alois P. Heinz, Table of n, a(n) for n = 0..313
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. A024451, A038603, A375534, A082830.

a[n_]:=Numerator[Sum[ 1/Part[ResourceFunction["OEISSequence"]["A038603"],i],{i,n}]]; Array[a,20] (* Stefano Spezia, Sep 06 2024 *)

a(0) prepended by Alois P. Heinz, Oct 21 2024

A179954 Decimal expansion of the sum of the reciprocals of pandigital numbers in which each digit appears exactly once.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 03 2010

This is example in 3. 1(a) of R. Baillie, revised.

This is a finite sum so it is a rational number.

			0.0008258903479192529386079577501789135432537929965887385729771528345968177...

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.
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series
Eric Weisstein's World of Mathematics, Digit
Wikipedia, Kempner series

Cf. A050278, A010784, A082830-A082839, A140502, A160502.

Sum_{k=1..3265920} 1/A050278(k).

Standardized offset and leading zeros from R. J. Mathar, Aug 06 2010

More terms from Robert G. Wilson v, Sep 07 2010

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

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

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A194181 Decimal expansion of the (finite) value of Sum_{k >= 1, k has no even digit in base 10 } 1/k.

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

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A375533 a(n) = numerator of Sum_{i=1..n} 1/A038603(i).

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A179954 Decimal expansion of the sum of the reciprocals of pandigital numbers in which each digit appears exactly once.

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Extensions

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

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