# [seqfan] Orderly Numbers

Andrew Weimholt andrew.weimholt at gmail.com
Fri Oct 30 12:58:40 CET 2009

```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...

1:  {1} == {1} mod 2
2:  {1,2} == {1,2} mod 3
5:  {1,5} == {1,2} mod 3
7:  {1,7} == {1,2} mod 5
8:  {1,2,8,4} == {1,2,3,4} mod 5
9:  {1,9,3} == {1,2,3} mod 7
11:  {1,11} == {1,2} mod 9
12:  {1,2,3,4,12,6} == {1,2,3,4,5,6} mod 7
13:  {1,13} == {1,2} mod 11
17:  {1,17} == {1,2} mod 15
19:  {1,19} == 1,2 mod 17
20:  {1,2,10,4,5,20} == {1,2,3,4,5,6} mod 7
23:  {1,23} == {1,2} mod 21
27:  {1,27,3,9} == {1,2,3,4} mod 5
29:  {1,29} == {1,2} mod 27
31:  {1,31} == {1,2} mod 29
37:  {1,37} == 1,2 mod 35
38:  {1,2,38,19} == {1,2,3,4} mod 5
41:  {1,41} == {1,2} mod 39
43:  {1,43} == {1,2} mod 41
47:  {1,47} == {1,2} mod 45
52:  {1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7
53:  {1,53} == {1,2} mod 51
57:  {1,57,3,19} == {1,2,3,4} mod 5
58:  {1,2,58,29} == {1,2,3,4} mod 5
59:  {1,59} == {1,2} mod 57
61:  {1,61} == {1,2} mod 59
67:  {1,67} == {1,2} mod 65
68:  {1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7
71:  {1,71} == {1,2} mod 69
72:  {1,2,3,4,18,6,72,8,9,36,24,12} == {1,2,3,4,5,6,7,8,9,10,11,12} mod 13
73:  {1,73} == {1,2} mod 71
76:  {1,2,38,4,19,76} == {1,2,3,4,5,6} mod 7
79:  {1,79} == {1,2} mod 77
83:  {1,83} == {1,2} mod 81
87:  {1,87,3,29} == {1,2,3,4} mod 5
89:  {1,89} == {1,2} mod 87
97:  {1,97} == {1,2} mod 95

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

```