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

%I A385489 #12 Jul 04 2025 20:02:15
%S A385489 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,
%T A385489 35,36,39,40,42,44,45,48,50,51,52,54,55,56,58,60,63,64,65,66,68,70,72,
%U A385489 74,75,76,78,80,81,82,84,85,87,88,90,91,95,96,98,99,100
%N A385489 Positive integers m such that every Gaussian integer g with |g| <= m is a linear combination of the distinct Gaussian divisors of m.
%C A385489 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.
%C A385489 It is assumed that all "semi-practical" numbers are members of this sequence.
%C A385489 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.
%H A385489 Frank M Jackson, <a href="/A385489/a385489_1.txt">Mathematica program that gives a complex plot</a>
%e A385489 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.
%e A385489 |= = = = = = = = = = = = + = = = = = = = = = = = =|
%e A385489 |            * * *     * * *     * * *            |
%e A385489 |        * * * * * * * * * * * * * * * * *        |
%e A385489 |    * * * * * * * * * * * * * * * * * * * * *    |
%e A385489 |    * * * * * * * * * * * * * * * * * * * * *    |
%e A385489 |  * * * * * * * * * * * * * * * * * * * * * * *  |
%e A385489 |  * * * * * * * * * * * * * * * * * * * * * * *  |
%e A385489 |* * * * * * * * * * * * * * * * * * * * * * * * *|
%e A385489 |* * * * * * * * * * * * @ * * * * * * * * * * * *|
%e A385489 |* * * * * * * * * @ @ @ @ @ @ @ * * * * * * * * *|
%e A385489 |  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
%e A385489 |  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
%e A385489 |* * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * * *|
%e A385489 +*-*-*-*-*-*-*-@-@-@-@-@-@-@-@-@-@-@-*-*-*-*-*-*-*+
%e A385489 |* * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * * *|
%e A385489 |  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
%e A385489 |  * * * * * * * @ @ @ @ @ @ @ @ @ * * * * * * *  |
%e A385489 |* * * * * * * * * @ @ @ @ @ @ @ * * * * * * * * *|
%e A385489 |* * * * * * * * * * * * @ * * * * * * * * * * * *|
%e A385489 |* * * * * * * * * * * * * * * * * * * * * * * * *|
%e A385489 |  * * * * * * * * * * * * * * * * * * * * * * *  |
%e A385489 |  * * * * * * * * * * * * * * * * * * * * * * *  |
%e A385489 |    * * * * * * * * * * * * * * * * * * * * *    |
%e A385489 |    * * * * * * * * * * * * * * * * * * * * *    |
%e A385489 |        * * * * * * * * * * * * * * * * *        |
%e A385489 |            * * *     * * *     * * *            |
%e A385489 |= = = = = = = = = = = = + = = = = = = = = = = = =|
%Y A385489 Cf. A005153, A363227.
%K A385489 nonn
%O A385489 1,2
%A A385489 _Frank M Jackson_, Jun 30 2025