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COMMENTS
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n: {divisors(n)} == {1,2,...,tau(n)} mod k
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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 3 or 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 3,5, or 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 3,7, or 21
27: {1,27,3,9} == {1,2,3,4} mod 5
29: {1,29} == {1,2} mod 3,9, or 27
31: {1,31} == {1,2} mod 29
37: {1,37} == 1,2 mod 5,7, or 35
38: {1,2,38,19} == {1,2,3,4} mod 5
41: {1,41} == {1,2} mod 3,13, or 39
43: {1,43} == {1,2} mod 41
47: {1,47} == {1,2} mod 3,5,9,15, or 45
52: {1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7
53: {1,53} == {1,2} mod 3,17, or 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 3,19, or 57
61: {1,61} == {1,2} mod 59
67: {1,67} == {1,2} mod 5,13, or 65
68: {1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7
71: {1,71} == {1,2} mod 3,23, or 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 7,11, or 77
83: {1,83} == {1,2} mod 3,9,27, or 81
87: {1,87,3,29} == {1,2,3,4} mod 5
89: {1,89} == {1,2} mod 3,29, or 87
97: {1,97} == {1,2} mod 5,19, or 95
The primes other than 3 are orderly.
Numbers of the form 4p are orderly when p is an odd prime congruent to 3,5, or 6 mod 7.
For primes, k values can be p-2 or a divisor of p-2 other than 1.
T. D. Noe observed that for composite orderly numbers, n, k seems to be one of the three values: tau(n)+1, tau(n)+3, tau(n)+4.
The composite numbers with k = tau(n)+4 are of the form p^2, where prime p == 3 mod 7.
The orderly numbers with k = tau(n)+3 come in many forms. See A168003. It appears that tau(n)+3 is a prime with primitive root 2 (A001122).
The forms for composite orderly numbers with k = tau(n)+1 are too numerous to list here, but seem to occur for any prime k > 3.
Let p be any prime. Then p^(m-2) is in this sequence if m is a prime with primitive root p. For example, 2^(m-2) is here for every m in A001122; 3^(m-2) is here for every m in A019334; 5^(m-2) is here for every m in A019335. For every prime p, there appear to be an infinite number of prime powers p^(m-2) here. All these numbers are actually very orderly (A167409) because we can choose k = tau(n)+1. - T. D. Noe, Nov 04 2009
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