A375524 -id:A375524 - cp's OEIS frontend

A375523 a(n) = numerator of Sum_{i=1..n} 1/A171397(i).

1, 3, 11, 25, 137, 49, 363, 761, 7129, 80939, 83249, 1109957, 1135697, 1159721, 2364487, 40916999, 13865893, 267536047, 271415923, 274943083, 6401288429, 6475652719, 32735212187, 33078431987, 300680459483, 43364113769, 1269032646901, 1280123549581, 40016557117411, 3666283538201

Offset: 1

N. J. A. Sloane, Aug 30 2024

Suggested by A375805.

			The first few sums are 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 80939/27720, 83249/27720, 1109957/360360, 1135697/360360, 1159721/360360, 2364487/720720,  ...

Alois P. Heinz, Table of n, a(n) for n = 1..2082

Cf. A171397, A375805, A375524, A001008/A002805.

b:= n-> (l-> add(l[i]*11^(i-1), i=1..nops(l)))(convert(n,base,10)):
g:= proc(n) option remember; `if`(n<1, 0, g(n-1)+1/b(n)) end:
a:= n-> numer(g(n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 30 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

cp's OEIS Frontend

A375523 a(n) = numerator of Sum_{i=1..n} 1/A171397(i).

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