[seqfan] Re: Arithmetic progressions such that adjacent terms have a common non-zero digit
Andrew Weimholt
andrew.weimholt at gmail.com
Tue May 28 20:07:49 CEST 2019
All numbers have the base-2 version of this property :-)
But it may be interesting to look at some other bases between 2 and 10.
Also might be worth looking at geometric progressions n, 2n, 4n, ... with
this property for various bases.
Andrew
On Sun, May 26, 2019 at 11:08 PM David Radcliffe <dradcliffe at gmail.com>
wrote:
> Daniel Griller asked the following question:
>
> Does there exist a positive integer n such that every term in the
> sequence n, 2n, 3n, 4n, 5n, ... has a non-zero digit in common with the
> next term?
>
> (Source: https://twitter.com/puzzlecritic/status/1125035277557354497)
>
> If we allowed 0 as a common digit then every multiple of 10 would be a
> solution.
>
> I was able to prove the following:
>
> - 99 is the smallest positive integer with this property.
> - 10^k - 1 has this property for every k > 1.
> - A positive integer n has this property if and only if 10*n has this
> property.
>
> Note that the adjacent terms k*n and (k+1)*n usually have the same leading
> digit, and it suffices to look at the boundaries where the leading digit
> changes.
>
> The sequence of numbers having this property might begin as follows, but I
> have not proved this.
>
> 99, 990, 999, 1998, 2997, 3996, 3999, 4992, 4995, 5994, 6875, 6993, 6996,
> 7992, 8125, 8704, 8991, 9856, 9900, 9984, 9990, 9999
>
> Is there an efficient algorithm to generate the terms of this sequence?
>
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