Newsgroups: sci.math, rec.puzzles
From: Dan Hoey <haoyuep@aol.com>
Date: Wed, 30 Mar 2011 16:48:41 -0400
Subject: Re: One Through Nine, With Eight Missing, And Two Ones

James Van Buskirk wrote:
> "Dan Hoey" <haoyuep@aol.com> wrote:
>> James Van Buskirk wrote:
>>> Given the O.P. I was hoping for something like the sum of the
>>> reciprocals of all the positive integers that don't have the digit
>>> '8'.

>> I like this more than Leroy Quet's puzzle.  Do you have any ideas
>> about how to calculate this number?

> I calculated up to double or quadruple precision once.  The version
> with 9 as the missing digit was from the Olympiad problem book and
> you can start with the proof that the series converges and apply
> a little more effort and get to something you can evaluate.  No one
> has ever given an independent calculation that I can check against,
> however.

According to http://oeis.org/A082837, the sum of reciprocals of
positive integers that don't have the digit 8 is
22.726365402679370602833644156742557889210702616360219843536376162.
For those without the digit 9, http://oeis.org/A082838 has
22.920676619264150348163657094375931914944762436998481568541998356.
Robert Baillie has a paper at http://arxiv.org/abs/0806.4410v2 that
covers these and related topics and includes Mathematica code.

Dan