An orderly number is an integer $n$ such that there exists at least one other integer $k$ such that each divisor^{} of $n$ from 1 to ${d}_{\tau (n)}$ (with $\tau (n)$ being the number of divisors function) satisfies the congruences^{} ${d}_{i}\equiv 1modk$, ${d}_{i}\equiv 2modk$ through ${d}_{i}\equiv \tau (n)modk$. For example, 20 is an orderly number, with $k=7$, since it has six divisors, 1, 2, 4, 5, 10, 20, and we can verify that

The orderly numbers less than 100 are 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, listed in A167408 of Neil Sloane’s OEIS. With the exception of 3, all prime numbers^{} are orderly.