{"status": "success", "data": {"description_md": "Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then $$f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\\cdots (p_k+1)^{e_k-1}.$$<br>For every $m\\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$s in the range $1\\le N\\le 400$ is the sequence $(f_1(N),f_2(N),f_3(N),\\ldots )$ unbounded?<br>'''Note:''' A sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$.\n\n$\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 17\\qquad\\textbf{(D)}\\ 18\\qquad\\textbf{(E)}\\ 19$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Define the function  <span class=\"katex--inline\">f_1</span>  on the positive integers by setting  <span class=\"katex--inline\">f_1(1)=1</span>  and if  <span class=\"katex--inline\">n=p_1^{e_1}p_2^{e_2}\\cdots p_k^{e_k}</span>  is the prime factorization of  <span class=\"katex--inline\">n&gt;1</span> , then  <span class=\"katex--display\">f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\\cdots (p_k+1)^{e_k-1}.</span> <br/>For every  <span class=\"katex--inline\">m\\ge 2</span> , let  <span class=\"katex--inline\">f_m(n)=f_1(f_{m-1}(n))</span> . For how many  <span class=\"katex--inline\">N</span> s in the range  <span class=\"katex--inline\">1\\le N\\le 400</span>  is the sequence  <span class=\"katex--inline\">(f_1(N),f_2(N),f_3(N),\\ldots )</span>  unbounded?<br/>&#8217;&#8217;&#8216;Note:&#8217;&#8217;&#8217; A sequence of positive numbers is unbounded if for every integer  <span class=\"katex--inline\">B</span> , there is a member of the sequence greater than  <span class=\"katex--inline\">B</span> .</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 15\\qquad\\textbf{(B)}\\ 16\\qquad\\textbf{(C)}\\ 17\\qquad\\textbf{(D)}\\ 18\\qquad\\textbf{(E)}\\ 19</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2012 AMC 12B Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12B_p25", "prev": "/problem/12_amc12B_p23"}}