Special forms and the distribution of practical numbers
Tóm tắt
A positive integer n is called practical if every positive integer $$m \leq n$$ can be written as a sum of distinct divisors of n. For any integers $$a, b, k > 0$$, we show that if $$2 \nmid a$$, then there are infinitely many nonnegative integers m such that $$am^{k} + bm^{k-1}$$ is practical. Let qn denote the n-th practical number. Further, when $$n \geq 7$$, we prove that $$\sqrt{q_{n}+1} - \sqrt{q_n} < \frac{1}{2}
$$ and there are at least two practical numbers between n2 and (n + 1)2.
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