LAUSR

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