A385489 Positive integers m such that every Gaussian integer g with |g| <= m is a linear combination of the distinct Gaussian divisors of m.

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 30, 32, 34, 35, 36, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 55, 56, 58, 60, 63, 64, 65, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 95, 96, 98, 99, 100

Offset: 1

Author

Frank M Jackson, Jun 30 2025

Keywords

• nonn

Comments

Practical numbers (A005153) are defined over Z+. A generalization of practical numbers over Z are known as "semi-practical" numbers (A363227). This sequence is a further generalization over the Gaussian integers.

It is assumed that all "semi-practical" numbers are members of this sequence.

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

Examples

			a(5) is in the sequence because the Gaussian divisors of 5 are 1, 1+2i, 2+i, 5. Each divisor has 3 other associates. In total these 16 divisors will give the complex plot below when they are combined linearly and distinctly. 5 is not a "semi-practical" number. Note also that every similar complex plot will give a pattern with the same number of axes of symmetry as that of a square.
|= = = = = = = = = = = = + = = = = = = = = = = = =|
|            * * *     * * *     * * *            |
|        * * * * * * * * * * * * * * * * *        |
|    * * * * * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * * * * * *    |
|  * * * * * * * * * * * * * * * * * * * * * * *  |
|  * * * * * * * * * * * * * * * * * * * * * * *  |
|* * * * * * * * * * * * * * * * * * * * * * * * *|
|* * * * * * * * * * * * @ * * * * * * * * * * * *|
|* * * * * * * * * @ @ @ @ @ @ @ * * * * * * * * *|
|  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
|  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
|* * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * * *|
+*-*-*-*-*-*-*-@-@-@-@-@-@-@-@-@-@-@-*-*-*-*-*-*-*+
|* * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * * *|
|  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
|  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
|* * * * * * * * * @ @ @ @ @ @ @ * * * * * * * * *|
|* * * * * * * * * * * * @ * * * * * * * * * * * *|
|* * * * * * * * * * * * * * * * * * * * * * * * *|
|  * * * * * * * * * * * * * * * * * * * * * * *  |
|  * * * * * * * * * * * * * * * * * * * * * * *  |
|    * * * * * * * * * * * * * * * * * * * * *    |
|    * * * * * * * * * * * * * * * * * * * * *    |
|        * * * * * * * * * * * * * * * * *        |
|            * * *     * * *     * * *            |
|= = = = = = = = = = = = + = = = = = = = = = = = =|
		

Links

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

Crossrefs

Cf. A005153, A363227.