
COMMENTS

. n: {divisors(n)} == {1,2,...,tau(n)} mod k
. 
. 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 p2 or a divisor of p2 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^(m2) is in this sequence if m is a prime with primitive root p. For example, 2^(m2) is here for every m in A001122; 3^(m2) is here for every m in A019334; 5^(m2) is here for every m in A019335. For every prime p, there appear to be an infinite number of prime powers p^(m2) here. All these numbers are actually very orderly (A167409) because we can choose k = tau(n)+1. [From T. D. Noe, Nov 04 2009]
