cp's OEIS Frontend

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).

%I A385916 #12 Jul 17 2025 15:59:55
%S A385916 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,
%T A385916 73,75,81,82,83,89,91,93,98,99,105
%N 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).
%C A385916 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|.
%C A385916 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.
%C A385916 An analogous sequence such that positive integers m that form the Gaussian integers m + i are prime is given by A005574.
%H A385916 Frank M Jackson, <a href="/A385916/a385916.txt">Mathematica program that gives a complex plot</a>
%e A385916 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.
%e A385916 |= = = = = = = = = = + = = = = = = = = = =|
%e A385916 |                    * * *                |
%e A385916 |              * * * * * * * *            |
%e A385916 |        * * * * * * * * * * * * *        |
%e A385916 |      * * * * * * * * * * * * * * *      |
%e A385916 |    * * * * * * * * * * * * * * * * *    |
%e A385916 |    * * * * * * * * * * * * * * * * *    |
%e A385916 |  * * * * * * * * * * * * * * * * * *    |
%e A385916 |  * * * * * * * * @ @ @ * * * * * * * *  |
%e A385916 |* * * * * * * * @ @ @ @ @ * * * * * * *  |
%e A385916 |* * * * * * * @ @ @ @ @ @ @ * * * * * *  |
%e A385916 +*-*-*-*-*-*-*-@-@-@-@-@-@-@-*-*-*-*-*-*-*+
%e A385916 |  * * * * * * @ @ @ @ @ @ @ * * * * * * *|
%e A385916 |  * * * * * * * @ @ @ @ @ * * * * * * * *|
%e A385916 |    * * * * * * * @ @ @ * * * * * * * *  |
%e A385916 |    * * * * * * * * * * * * * * * * * *  |
%e A385916 |    * * * * * * * * * * * * * * * * *    |
%e A385916 |    * * * * * * * * * * * * * * * * *    |
%e A385916 |      * * * * * * * * * * * * * * *      |
%e A385916 |        * * * * * * * * * * * * *        |
%e A385916 |            * * * * * * * *              |
%e A385916 |                * * *                    |
%e A385916 |= = = = = = = = = = + = = = = = = = = = =|
%Y A385916 Cf. A005574, A005153, A363227, A385489.
%K A385916 nonn,more
%O A385916 1,2
%A A385916 _Frank M Jackson_, Jul 12 2025