# [seqfan] Re: Orderly Numbers

Alonso Del Arte alonso.delarte at gmail.com
Fri Oct 30 16:34:37 CET 2009

```A potentially dumb question, but: how do you choose the k for each n? Are
there "orderly" numbers with only one choice for k and others with lots of
choices?

Al

On Fri, Oct 30, 2009 at 7:58 AM, Andrew Weimholt
<andrew.weimholt at gmail.com>wrote:

> I am tentatively using the term "orderly numbers" to indicate numbers, n,
> for which there exists some number, k > tau(n), such that the set of
> divisors of n is congruent to the set {1,2,...tau(n)} mod k.
>
> Numbers without this property can be called "disorderly numbers".
>
> For example, 12 is orderly, because the divisors of 12 are 1,2,3,4,6,12,
> and
> 1 == 1 mod 7
> 2 == 2 mod 7
> 3 == 3 mod 7
> 4 == 4 mod 7
> 12 == 5 mod 7
> 6 == 6 mod 7
>
> I will use the more compact notation 12: {1,2,3,4,12,6} == {1,2,3,4,5,6}
> mod 7
> to list the orderly numbers under 100 below...
>    ...

The Orderly Numbers...
> 1,2,5,7,8,9,11,12,13,17,19,20,23,27,29,31,37,38,41,43,47,52,53,57,58,59,
> 61,67,68,71,72,73,76,79,83,87,89,97,...
>
> The Disorderly Numbers...
> 3,4,6,10,14,15,16,18,21,22,24,25,26,28,30,32,33,34,35,36,39,40,42,44,45,
> 46,48,49,50,51,54,55,56,60,62,63,64,65,66,69,70,74,75,77,78,80,81,82,84,
> 85,86,88,90,91,92,93,94,95,96,98,99,100,...
>
> (not sure what to do with 0. My instinct is to leave it out, and apply the
> terms "orderly" and "disorderly" only to the naturals.)
>
> The primes, except for 3, are orderly.
> Numbers of the form 4p (p prime) are orderly when (but not necessarily
> only when) p == 3,5, or 6 mod 7.
>
> Those of you who find this post interesting may also be interested
> in a related sequence that I submitted last year: A140539
>
> Andrew
>
>
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>

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