Directed graphs of integers with arcs determined by an arithmetic function

<p>We introduce a new arc in directed graphs of integers. Our arcs are determined by the values of the popular arithmetic functions such as the divisor function <span class="math inline">\(\tau\)</span>, the prime divisors counting functions <span class="math inline">\(\omega\)</span> and <span class="math inline">\(\Omega\)</span>, and the sum of digits function <span class="math inline">\(s_b\)</span>, evaluated at the multiples <span class="math inline">\(N\)</span> of a particular integer. Among other things, we determine the positive integers that have arcs to all except a finite number of positive integers. We also propose some possible research problems at the end of this article.</p>