A082830 - OEIS

%I #35 Feb 16 2025 08:32:49

%S 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,

%T 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,

%U 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

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

%C Such sums are called Kempner series, see A082839 (the analog for digit 0) for more information. - _M. F. Hasler_, Jan 13 2020

%D Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

%H Robert Baillie, <a href="http://www.jstor.org/stable/2321096">Sums of reciprocals of integers missing a given digit</a>, Amer. Math. Monthly, 86 (1979), 372-374.

%H Robert Baillie, <a href="http://arxiv.org/abs/0806.4410">Summing the curious series of Kempner and Irwin</a>, arXiv:0806.4410 [math.CA], 2008-2015. [From _Robert G. Wilson v_, Jun 01 2009]

%H Eric Weisstein's World of Mathematics,, <a href="https://mathworld.wolfram.com/KempnerSeries.html">Kempner Series</a>. [From _R. J. Mathar_, Aug 07 2010]

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Kempner_series">Kempner series</a>.

%H Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, <a href="http://library.wolfram.com/infocenter/MathSource/7166/"> Summing Kempner's Curious (Slowly-Convergent) Series</a>. [From _Robert G. Wilson v_, Jun 01 2009]

%F 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

%e 16.17696952812344426657...

%t (* see the Mmca in Wolfram Library Archive. - _Robert G. Wilson v_, Jun 01 2009 *)

%Y Cf. A002387, A024101, A052383 (numbers without '1'), A011531 (numbers with '1').

%Y Cf. A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 2, ..., 9 and 0).

%K nonn,cons,base

%O 2,2

%A _Robert G. Wilson v_, Apr 14 2003

%E More terms from _Robert G. Wilson v_, Jun 01 2009