A385916 Positive integers m that form Gaussian integers m + i such that every Gaussian integer g with |g| <= |m + i| is a linear combination of the distinct Gaussian divisors of m + i (where i is the imaginary unit).

1, 2, 3, 5, 7, 8, 12, 13, 17, 18, 21, 23, 27, 31, 32, 33, 37, 38, 41, 43, 47, 55, 57, 68, 72, 73, 75, 81, 82, 83, 89, 91, 93, 98, 99, 105

Offset: 1

Author

Frank M Jackson, Jul 12 2025

Keywords

• nonn
• more

Comments

This sequence is an example of extending the concept of a practical number to the domain of Gaussian integers. To determine if a Gaussian integer p is practical over the Gaussian integer domain it is necessary to show that the Gaussian divisors (including all their associates) of the Gaussian integer p when combined linearly and distinctly generate all Gaussian integers g where |g| <= |p|.

The Mathematica program in the link below gives a complex plot of the linear combinations of the distinct divisors of a Gaussian integer m + i to see if it is a member of this sequence.

An analogous sequence such that positive integers m that form the Gaussian integers m + i are prime is given by A005574.

Examples

			a(3) is in the sequence because the Gaussian divisors of 3 + i are 1, 1 + i, 1 + 2i, 3 + i. Each divisor has 3 other associates. In total these 16 divisors will give the complex plot below when they are combined linearly and distinctly. Note that the patten in any quadrant is a rotation by a right angle of its adjacent quadrant.
|= = = = = = = = = = + = = = = = = = = = =|
|                    * * *                |
|              * * * * * * * *            |
|        * * * * * * * * * * * * *        |
|      * * * * * * * * * * * * * * *      |
|    * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * *    |
|  * * * * * * * * * * * * * * * * * *    |
|  * * * * * * * * @ @ @ * * * * * * * *  |
|* * * * * * * * @ @ @ @ @ * * * * * * *  |
|* * * * * * * @ @ @ @ @ @ @ * * * * * *  |
+*-*-*-*-*-*-*-@-@-@-@-@-@-@-*-*-*-*-*-*-*+
|  * * * * * * @ @ @ @ @ @ @ * * * * * * *|
|  * * * * * * * @ @ @ @ @ * * * * * * * *|
|    * * * * * * * @ @ @ * * * * * * * *  |
|    * * * * * * * * * * * * * * * * * *  |
|    * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * *    |
|      * * * * * * * * * * * * * * *      |
|        * * * * * * * * * * * * *        |
|            * * * * * * * *              |
|                * * *                    |
|= = = = = = = = = = + = = = = = = = = = =|
		

Links

• Frank M Jackson, Mathematica program that gives a complex plot

Crossrefs

Cf. A005574, A005153, A363227, A385489.