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
Canonical name	OrderlyNumber
Date of creation	2013-03-22 19:05:28
Last modified on	2013-03-22 19:05:28
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A07