orderly number


An orderly number is an integer n such that there exists at least one other integer k such that each divisorMathworldPlanetmathPlanetmath of n from 1 to dτ(n) (with τ(n) being the number of divisors function) satisfies the congruencesMathworldPlanetmathPlanetmathPlanetmathPlanetmath di1modk, di2modk through diτ(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

11mod7
22mod7
103mod7
44mod7
55mod7
206mod7

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 numbersMathworldPlanetmath 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