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

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

2,1

Comments

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

References

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.

Links

Table of n, a(n) for n=2..49.

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]

Example

26.2833282048814207699401516874442229241887980925...

Crossrefs

Cf. A171397, A375523/A375524.

See also A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839.

Sequence in context: A091944 A069936 A043294 * A365260 A368522 A033641

Adjacent sequences: A375802 A375803 A375804 * A375806 A375807 A375808

Keyword

nonn,cons,base,more

Author

Robert C. Lyons, Aug 29 2024

Extensions

Corrected data provided by Gareth McCaughan, Sep 02 2024

Status

approved