{"status": "success", "data": {"description_md": "**POTD December 9, 2023**\n**Author:** munch\n\nFind the number of values of $n$, such that $n \\in \\mathbb{N}$ and $n > 1$, for which there exists a set of positive integers $\\{a_1, a_2, ..., a_{n-1}\\}$, which satisfy the following conditions:\n1. For $1 \\leq i \\leq n-1, a_i > i$ and $n \\nmid a_i$\n2. For $1 \\leq i < j \\leq n-1, a_i \\text{ is not} \\equiv a_j (\\bmod n)$.\n3. There exists $k$, such that $k \\in \\mathbb{Z}^+, k > 1$ and $1+k^{a_1}+k^{a_2}+...+k^{a_{n-1}}$ is prime.", "description_html": "<p><strong>POTD December 9, 2023</strong><br/>&#10;<strong>Author:</strong> munch</p>&#10;<p>Find the number of values of <span class=\"katex--inline\">n</span>, such that <span class=\"katex--inline\">n \\in \\mathbb{N}</span> and <span class=\"katex--inline\">n &gt; 1</span>, for which there exists a set of positive integers <span class=\"katex--inline\">\\{a_1, a_2, ..., a_{n-1}\\}</span>, which satisfy the following conditions:</p>&#10;<ol>&#10;<li>For <span class=\"katex--inline\">1 \\leq i \\leq n-1, a_i &gt; i</span> and <span class=\"katex--inline\">n \\nmid a_i</span></li>&#10;<li>For <span class=\"katex--inline\">1 \\leq i &lt; j \\leq n-1, a_i \\text{ is not} \\equiv a_j (\\bmod n)</span>.</li>&#10;<li>There exists <span class=\"katex--inline\">k</span>, such that <span class=\"katex--inline\">k \\in \\mathbb{Z}^+, k &gt; 1</span> and <span class=\"katex--inline\">1+k^{a_1}+k^{a_2}+...+k^{a_{n-1}}</span> is prime.</li>&#10;</ol>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "Problem of the Day #20", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}