Special forms and the distribution of practical numbers (Q2179854)

\textit{A. K. Srinivasan} [Practical numbers. Curr. Sci. 6, 179--180 (1948)] introduced the notion of practical numbers (a positive integer \(n\) is said to be practical if every positive integer \(m\leqslant n\) can be written as a sum of distinct divisors of \(n\)) and showed that a practical number greater than \(2\) is divisible by \(4\) or \(6\). \textit{B. M. Stewart} [Am. J. Math. 76, 779--785 (1954). Zbl 0056.27004] gave a characterization of practical numbers in terms of their prime factorization, and this is used here to show two results on practical numbers as follows. For positive integers \(a,b,k\) with \(a\) odd, it is shown that \(am^k+bm^{k-1}\) is practical for infinitely many \(m\geqslant1\). The second result shows that for \(n\geqslant7\) there are at least two practical numbers in the interval \((n^2,(n+1)^2)\). It is also conjectured that for any \(k\) this interval contains at least \(k\) practical numbers for \(n\) large enough.