# orderly number

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 $d_{i}\equiv 1\mod k$, $d_{i}\equiv 2\mod k$ through $d_{i}\equiv\tau(n)\mod k$. 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

 $1\equiv 1\mod 7$
 $2\equiv 2\mod 7$
 $10\equiv 3\mod 7$
 $4\equiv 4\mod 7$
 $5\equiv 5\mod 7$
 $20\equiv 6\mod 7$

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.

Title orderly number OrderlyNumber 2013-03-22 19:05:28 2013-03-22 19:05:28 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A07