论文标题
关于实际数字的一些结果
On Some Results on Practical Numbers
论文作者
论文摘要
如果可以将$ [1，n] $中的每个整数表示为$ n $的不同除数的总和，则据说一个正整数$ n $是一个实用的数字。在本文中，我们考虑了给定多项式形式的实际数量。我们为系数$ a $ a $ a $ a $ a $ a $ a+b $提供了必要和充分的条件。我们还为二次多项式提供了必要的足够和足够的数字，使用了[9]中提到的猜想的第一部分。在最后一部分中，我们证明，每一个$ 8K+1 $的表格都可以表示为实用数字和一个正方形的总和，并且每$ j \ in \ {0，\ ldots，7 \} \ setMinus \ {1 \ {1 \} $都有无限的自然人数量$ 8K+j $的形式，该$ 8K+j $不可能按正方形和一个正方形的数字和一个正方形的数字。
A positive integer $n$ is said to be a practical number if every integer in $[1,n]$ can be represented as the sum of distinct divisors of $n$. In this article, we consider practical numbers of a given polynomial form. We give a necessary and sufficient condition on coefficients $a$ and $b$ for there to be infinitely many practical numbers of the form $an+b$. We also give a necessary and sufficient for a quadratic polynomial to contain infinitely many practical numbers, using which we solve first part of a conjecture mentioned in [9]. In the final section, we prove that every number of $8k+1$ form can be expressed as a sum of a practical number and a square, and for every $j\in \{0,\ldots,7\}\setminus \{1\}$ there are infinitely many natural numbers of $8k+j$ form which cannot be written as sum of a square and a practical number.
