Problem 53735. Easy Sequences 58: Curious Prime-Rational Functions

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For some prime numbers </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>p</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>q</w:t></w:r></w:customXml><w:r><w:t> where </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>p &lt; q</w:t></w:r></w:customXml><w:r><w:t>, a rational function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R</w:t></w:r></w:customXml><w:r><w:t>, is defined as follows: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R(p,q)=\\frac{2pq+p+q+1}{pq+q} </w:t></w:r></w:customXml><w:r><w:t>. Using the output </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>r = R(p,q)</w:t></w:r></w:customXml><w:r><w:t>, another rational function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t>, is defined: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>K(r)\\ =\\ \\frac{2r}{2-r} </w:t></w:r></w:customXml><w:r><w:t>. Finaly, using the output </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k=K(r)</w:t></w:r></w:customXml><w:r><w:t>, we define the function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>N</w:t></w:r></w:customXml><w:r><w:t>: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>N(k)=\\begin{cases}k&amp;\\left\\lfloor k\\right\\rfloor =k\\\\ 0&amp;\\left\\lfloor k\\right\\rfloor \\neq k\\end{cases} </w:t></w:r></w:customXml><w:r><w:t>; where the symbol \"</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\left\\lfloor \\ \\right\\rfloor </w:t></w:r></w:customXml><w:r><w:t>\", represents the integer part of the decimal expansion of the fraction </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t> .</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>p=2</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>q=5</w:t></w:r></w:customXml><w:r><w:t>: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>r=R(2,5)= \\frac_{28}_{15}</w:t></w:r></w:customXml><w:r><w:t>; </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k=K(\\frac_{28}_{15})=28</w:t></w:r></w:customXml><w:r><w:t>; since </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\left\\lfloor k\\right\\rfloor = k\\end</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\therefore N(k)=28</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>And, for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>p=3</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>q=7</w:t></w:r></w:customXml><w:r><w:t>: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>r=R(3,7)= \\frac_{53}_{28}</w:t></w:r></w:customXml><w:r><w:t>; </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k=K(\\frac_{53}_{28})= \\frac_{106}_{3}</w:t></w:r></w:customXml><w:r><w:t>; since</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\left\\lfloor k\\right\\rfloor \\neq k\\end</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\therefore N(k)=0</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>If </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n=N(k)</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, and given an integer limit </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, write a function that returns a sorted array of all </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>unique</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> that are less than or equal to </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>-----------</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>HINT: </w:t></w:r><w:r><w:t>Both </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t>, are rational functions and expect </w:t></w:r><w:r><w:rPr><w:u/></w:rPr><w:t>exact</w:t></w:r><w:r><w:t> rational fraction outputs. Therefore, please preserve numerators and denominators for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t>, and evaluate decimal expansions only when calculating the output of the function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>N</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 357.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 178.75px; transform-origin: 407px 178.75px; vertical-align: baseline; "><div style="block-size: 130.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 65.25px; text-align: left; transform-origin: 384px 65.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81px 8px; transform-origin: 81px 8px; unicode-bidi: normal; ">For some prime numbers p<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and q<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; "> where <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEwAAAAkCAYAAAAjMNwcAAACEElEQVRoge2ZcbGDMAyHPw91gIEZQMEU4GAOcIAFNEwCHrAwDbPA+6PNI+wVGC2FvV2/O+526y1Jfw1JYJDJZDKZTCaTyWQyn4YBirOD+A8YoAaewO3kWD4aLdTgruupEc1ggBKomJ7oFbuB2q2n9P8qVMO+t6PsMWofBTbQB2OgrTP6AHrPmolxOOM/pVAl0DkfDXYPrfvcAfdQoxLwHSuSDlqLtkemFdigB9IJhfJxZ3rQLeMhtSGGb4yB9/zNolqt1yEOHEcJZbAiiViv6P0E1UgxPgCXFQdNgP2jhBLE12NlfQh1ID/uZta1YNUGu0cLBTZjlg4fxhITXb/mZp7ujSB8XJnWvwdpxwTDuhjv7HeRRhnwnbxR632IA2xWHiFcpXzMNSd9+EGZ3rN8O+qGELvJ1MLpWuxDZ1fQ4evs8XW/d1I8hFTCyajgE0PvJbjb6wLpS2HdbfYcWAUZKvcSbi57jPtO+wqaJ3UHexVE6kEqsTR7CaczSJpTgRVLJvwBm4lBaAdSAA1jI4gZUkPwCbelK+t6+2SszzI7Rk33hTLeOaO1M1aRPquWKBmzYWvrr7D1tsPuRxJBl58ts+Qv+jRSvomIoWC/2KKn+7UW/G3I7Tg3PqU38I+4EDnd6wHu6MJ+BvppZksTAWwx153o2wUrmb6g3PQ4JG84fde3/TtTYBPDd4W8ospkMsfzA32t/Hm22hXTAAAAAElFTkSuQmCC" style="width: 38px; height: 18px;" width="38" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.5px 8px; transform-origin: 62.5px 8px; unicode-bidi: normal; ">, a rational function R<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71px 8px; transform-origin: 71px 8px; unicode-bidi: normal; ">, is defined as follows: <img src="data:image/png;base64,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" style="width: 165px; height: 35px;" width="165" height="35"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8px 8px; transform-origin: 8px 8px; unicode-bidi: normal; ">. Using the output <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJEAAAAlCAYAAABCgwtAAAAEGElEQVR4nO1b0ZHqMAzcHtJBGqCBVHAVpAM6SAe0kBooIT3QAjXQwr0Po8nG2LFs514gaGcyN3dwwYjVaiUbwGAwGAwGg8Fg2A5nAM3ei9gAPY7xPj4OI1zwj4ATgOn50+ChAdAprjbzviOcCh0JJwB3GJFe0MCpxQ3Ar3fd4LKPf9fI+vD8vyOig4uDlbYAOiwJ9EOPtViSbEQ8iHKfXOX6JFwBXPZexDvCJ5GPFnGSMSY4kh0ZEotu74W8Gy6YCRIrRaxGoUz8BhUSTHCKZCAwQWKGmP1RSG2uz/t8A3p8T8Ko0GBZqmLdxwNxosk9vsUrSEmr7kD91veEz6yTP5jJ8Yg85+w9xzfWco+YV1pDCxe3M/2/dI3D8+9/3VbLKCOn63qgoKSd4ST9jjmgzfNiqR8y7qmZ0aSu2gCPWC9TPdJKNSDfbHZwysUK18PF+QEX0zX1q0WL+b2Pz7VMmD/nCevlSrhQBPEPMi+4YUmsHEn3ZzMlV63B89cu5Byw9Eo3xAl7Rbir00AIKO/lilkRuCsMKWApJDHuWBKlx0zcFEFEOIoggb1gOTOQ33PM1rTBlaN8PvzW/YYlcR5wmZl6TzUBFQLGygOr/BZ2QUh7xyspedSREgO5T7a5ZhN6wed3I+x1pLX3iaUJUg2JJPNjfoxJVOK5GFyaQ++LH08RtqSEA1ia0CPso4gK+F4u9vcYSknEmR97HSZRTUvdYiZszF+JR4oRmlFMIjahRxg2sXHlYHBGavaKSknEfiiWkKnOUQsumyE0mOOh+WxlQJvt09iEbqFCe3ZnJyzLlo8YwUKQ5MqF+K+YiWXlr5lBcYmOKR4TWnOEpShxOOhbqdCe3RkHLbTVkWr9Q/fKkXb2l7EPlstqTWfG3i+0RlYh7WtNKPDEvJBag8cL2as7Y68R8gicNKFOhiHeJucQGqtM6IPl16+dEaXIyI9rj7FokusFEvTa2vwO0G51cPlOEeSBvJLDm76h9UmpiymtzLI0Z5w4Yfznjt7jmqQUgmed3OSgH+GoA6vAmnyz+qbK5oi8Ca4QNFQSpJTGXtMfQ2jW5qtaQ2vmRkLjMQcUDD856FuVsr3AWZ5qnbXniIC5pGl8Ed+Xu78W83bH2mv5TUGq/fefP9HPBnM50ybBHQViwvspn4weLmC+r7oiLuMX6D2Y9lAaZ77sW8mlTVI5hisJkZohdZhLl2zvAPlVpoMjuR0D+SPIQC+lRmxka9GhYiMUeVNqwJG2ZqvJoIAc/F/zC9JOb6HsV9R1bzmEPvKXEN4OA+Klgf1JbUZLKayBdkpt3/TYAT3CROIhZ82ufI/6L0dyw7SmZvLlRSPQDvBPC/ImaC2JauF/FWqNkJ/ekR8GMhwMXf8boQ5VOkvrugwGg8FgMBg2wj/ilRqXnMpvgAAAAABJRU5ErkJggg==" style="width: 72.5px; height: 18.5px;" width="72.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 83px 8px; transform-origin: 83px 8px; unicode-bidi: normal; ">, another rational function K<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.5px 8px; transform-origin: 38.5px 8px; unicode-bidi: normal; ">, is defined: <img src="data:image/png;base64,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" style="width: 94.5px; height: 35px;" width="94.5" height="35"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 83.5px 8px; transform-origin: 83.5px 8px; unicode-bidi: normal; ">. Finaly, using the output <img src="data:image/png;base64,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" style="width: 58.5px; height: 18.5px;" width="58.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, we define the function N<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">: <img src="data:image/png;base64,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" style="width: 129px; height: 39.5px;" width="129" height="39.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 65.5px 8px; transform-origin: 65.5px 8px; unicode-bidi: normal; ">; where the symbol "<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC8AAAAlCAYAAADSvLDKAAAAh0lEQVRYhe3YwQmDQBgF4enBDtKADViBFaQDO7CmlGAx9hIPelqSwMNfhDADy96eH3sUzC5pBB43fLOrGFqAuWIoaAWGiqE78G/Ei48TD+LjxIP4OPEgPk48iI8TD+LjxIP4OPEgPk48/AF+Pe72PE9uv77sluFH9pf/dPqT29OP7ZK/xGZNG+TqVZ2JawLPAAAAAElFTkSuQmCC" style="width: 23.5px; height: 18.5px;" width="23.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 139px 8px; transform-origin: 139px 8px; unicode-bidi: normal; ">", represents the integer part of the decimal expansion of the fraction k<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; "> .</div><div style="block-size: 28px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 14px; text-align: left; transform-origin: 384px 14px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.5px 8px; transform-origin: 51.5px 8px; unicode-bidi: normal; ">For example for <img src="data:image/png;base64,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" style="width: 38px; height: 18px;" width="38" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" style="width: 36.5px; height: 18px;" width="36.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">: <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMcAAAA4CAYAAABJwewKAAAFtklEQVR4nO2dW7HqMBiFlwcc1AAGUIACHNQBDrCABiTwfh6xgAYs7PMQ1vQnJG3Se8r6ZjJ7KKWX9L8n6QaEEEIIIYQQQgghhBBCCCFmYQ/gBuAF4A/AE8AxsF8F4ArgAuAM4PH+K8Qm2cMpQw3gACf4f+9mBX8HpzxHb9sTTmGE2Bw3fHuJGk45XoFtO2/fc2S7EMVzx7dg79B4D35HJdh7+17glEjKIX4G5h4U+kNgG0MthVXiZ6jgFOHmbb+iCbdqNAm5vIb4GU5wCuCHUECjIL4XERNRo9xOrpe+gAl4Ilyi3cHlKLbsG8pDxEhc4SxVqRwRTmpL5Qz3TEL3c4cLpXZoFIVh1lbuP5sdXELW1arM416RZnnt+XPPkUuF+P3VCCvyCY3QlEyboh/hFME+rx3cffvbf4odGgH489oDjQXh5xO6BeX8/l0be+/YNtY99LiPFELnY2uzkLS4pcK+jhkflnL9MREqzWW6SysDlvPYbEdV+FSemGu2x2nzAuz0thaa4jAE//781jZVgqPFY1/THHCUPPQ8+BxPCHuIQ2T7z+ELj0+FNOG9o93Ksn5+x6eHoNu35xgzlGGy2RdWeUoKryo0fX322h2NR2AI9fB+f3tvmzrcXT123k0sJLLeI+RqU7zGBe0xvFWQsSzWHsO9EZW6FCtqc4ZY23v7X9GMb1zf7ecVA/jsyJgAWMENeQdamjZiNXZiQ64hlt7C+v3QpP8KF6KIH8LOt2mrbdv6t69APEZb8sZp0V2MqRx+OMik/4x8RWFsrtp/Ar4l2mO6SsuUWGv9iuxTe/v4YRGPMUbSmpIkp2JHfUMtJ0xieFZKaDUrNVxo8cRn0mgHTXIfasoYQ1cbasmsAIUs+wmfAhU6H8uBQ42DtfRjWWgaLS7i8RUkp1QZ66MuUseTxh5vmh12MBPLBz4VJrezh7ah4Yd/7XwQvjA9EBfYG8JVrlyoiF3jJEM44FtJUpW677V1lZJT2+q9Fjv2AicUVAZ+ztHu+whtSPjhx+Qs6dkQqk64J3rOofA4c8T11mOmCjwjh1w4EDe0rXqsxSavLEuWjM0lKCC+wqQo+xjKQes659plGoLUMYyxjMAmscnrE+VXLhgO+UIZ2x5jqNAstWbZ5lMpz7IU5fiHccK4rLDTuuKx6vBLYsuztgOs0KRMvBsqNDcsMwPWRgIpyhEaSV4jiyiHTV7H8BpLVqtYmmTziSlOCBqNPlyx7NTwnPCxb0K++WqVFaaxvMYYWt33Wlh+jT3wrhJv6Fi5pdylFYOeIzXJZq6Zy+arVTZ5HatqsGS1qmsekzUGXcsm+fBzFjdd0K0Yh47vh8LwMUXg+9wj2Xy1yq6OKp3UKSM2jOwSihfSreoJ8anUZI/v+P4EZwxShaQr7HwmXAehcVxtaLMUVphKXvhC/DUVMetsvWVX+JY6MY/W+obvqdRsfGfSOfC71BDDVtxC409XfL/prw1a719ijwRjYIVpta4tkdB05lgHpK7jAJqwoy3v8AW8rb2866q977sU0eZUPB7fBfuEE/TUYgb7oU9ItQYY1rV59tCz6QqnATTzqkq3HCc0ZVPbaMVDXLx923KcO+KetQqct635D5JrDjhKneKlDuY39j5zCwecSlMaFdy1s/LYphzsc+vBSzUGq4Srz6aenTznuu657mkKKjijQi8aU44D9K6qWThh2jd2cDR9rtkJQ+evrYEu5biheSN76bM+Vs9Ulp350lx5X+lvHiFtyhFaEKa14xNzwviC1Sdf6MsU178UKWHVEd+LwxRmTcjUg3hTUnpV0tKlHBb7KqYtzB8UopUc5QCaIkTf+XJCFEOucgBNiFWq5xciiT7KoX97Jn6CPsqxhZWuQnTSRzk47iHEpuHrXkPKEZpawteBKqQSm6WCs/5UgBfc+I0d4ONETH7PfzokxRBCCCGEEEIIIYQQQogy+Q/yjAomtnSc3QAAAABJRU5ErkJggg==" style="width: 99.5px; height: 28px;" width="99.5" height="28"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">; <img src="data:image/png;base64,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" style="width: 95.5px; height: 28px;" width="95.5" height="28"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.5px 8px; transform-origin: 22.5px 8px; unicode-bidi: normal; ">; since <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGIAAAAlCAYAAAC5+DzaAAACbklEQVRoge1aYbGDMAz+POAAAxiYAhTgAAdzMAvTgAQ8zAIaZmHvR5trBuWtafsGuZfvrnejgybtlyZpADAYDAaV6AG0B8hsvizz9JgBXL8scwFw+bLM0+MIIl4wIjbIIaKDcy83AGOGTO1ENHD6jwAmVHKzEiJ6f/8TbjFfvk8KrUQ0AO4AHgjzf9QaPGdH3JkiOdaglQhCizD/W61Bc4iYvRJzpkztRFwQiKg2jxwiSIncIK+diCvCGlSDlIga1qCdiFKPsDuohIga1qCdiFKPEIWUCMoYSqxBMxE9/iA+ADIiGqYEnR9a/zyltA98LmGUTqL1z5e2nIzv5vV/sr4B7jzBd0onHVh6jiBhXFDjFUsdp5QI7h5LmnixEDzCFNFpQUHdTkJEzBoar5xkUqVEDHB6lzbposU8AumzJkYMCREL3q2h9b+lE9IaIwZsd9OISkE7lQh+mhzhFnLB/zpZU0XhiVDuqJY5pRLBrYF89H8jgnsEihU5Rc8oUongWQEv+h1RfT0ia+rwHuhpDXKNcYNUIkjwjPegRdcSaMyaRvbcBSFxya1Ab5BCBLeGwffxCqx0UTVmTeQRFn/NY2aVUkcKEdwaSHlOzrS699Mia4wR5BHurI+7a9pdzeqeZKQQQUWu9UsQ6qft2SFYzG/QRgQvdPY7/eSiJwSvIUIKEXsvQVoES3n6luJ3tRHBY9I6HvJYsaDgYPeJiA7Bp8YWufPCp53/Y9BGxA1u/ntv466o8BGGfcVxEhgRJ4ERcRIYESeBEXESzHBpV+zkmZUPM0w74xoREfRwOyLWct5gcVCtPtbsa3CDwWA4Fj+ahkUsC30dVAAAAABJRU5ErkJggg==" style="width: 49px; height: 18.5px;" width="49" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <img src="data:image/png;base64,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" style="width: 81px; height: 18.5px;" width="81" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</div><div style="block-size: 28px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 14px; text-align: left; transform-origin: 384px 14px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27px 8px; transform-origin: 27px 8px; unicode-bidi: normal; ">And, for <img src="data:image/png;base64,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" style="width: 38px; height: 18px;" width="38" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAkCAYAAADFGRdYAAABoElEQVRoge2Y8bGDIAzGvx3YwAVcwAk6QTdwAzfoCp2hI7CDKzBDV/D9gTlzffIMoE+o+d1x1xMaIOYjQUBRFEVRlDppAHQR7ZI8AUzCNp60xlMxkDtoAjCcs8xzGQA4bMtohHdSc/iKCsQBaDfGNLi41G6CcT0uLDUpJLVdMhul0q3wrQmSmss1dJ+NOPiQtPAh+pp/S8PUIK5eCbU9D1eS2iPVgMESip+OoOcTvBMldIhLx6HWp25ohSyptczA2qIslkWbCJt2hyY5jCVkSY1H0DMwxs39NmWCQsiS2mP+8zvQT29g79D/b5Kl1mLbAQMbU2umy5IavwyuYeAjLGWCkrIbvegkqZEDQmcNj6LYCUrKbnSmZkltzQG8P2WCUrIb7SNJavxNfzqphY8ynvprhRJTKHP/Cc9aIxbtd/AOosp7gq+2a4X2kByRLyyOoqsIGeRyk1bZpUF7eENeBP/CwB+MFt5hPTNGYVrzx6k74u6b0XxDlX0oXGr6cSoA3XNqrrIPh+45ofvc5eFRNCIjK3wjN4Sr3lpLAEVRyuUH8PPjj4ANjYoAAAAASUVORK5CYII=" style="width: 36.5px; height: 18px;" width="36.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">: <img src="data:image/png;base64,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" style="width: 99.5px; height: 28px;" width="99.5" height="28"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">; <img src="data:image/png;base64,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" style="width: 95.5px; height: 28px;" width="95.5" height="28"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.5px 8px; transform-origin: 20.5px 8px; unicode-bidi: normal; ">; since<img src="data:image/png;base64,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" style="width: 49px; height: 18.5px;" width="49" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJMAAAAlCAYAAABGdtt9AAAD/UlEQVR4nO2bfZHCMBDFnwcc1EANVAEKcIADHGABDZWAh1pAQy1wf4Q3XXJNspsC13L7m+nMzVzalOTtZwBwHMdxHMdxnDx7AO0LntM8nuX8Uw4ALi983unFz3MW0s1cu8z4NnFPydscAPTK96HoOsX4C75LUA2mNd0cHYKF38V1zIxvETZ7FOPPCIuQm2NEXqQnANfoPXLjJdfHO22ZA4AbgsHR444Ie6Fdh9VwwbSJV5Q/wBmTkHLsEBZFu9m3x3MH5XggCHxEXtBrhmsfr+UR+v1YFVJMd5TDVq8cd0IQiBbOfzLcA4T3vxrvWQMUzA3zgqG3LhntqqBH4FXKQ0aURWL1Sp2Y35oz8N5XVIqfguuTE8sBegMvTkZl9nifq2sweQO+eC7HaaGzFlqd9r3l/DXcsK1knOtzR7rN0Ygxi7yTnOyO9yWZnKfFc7hLJeIcX/IeA2y5Dw2nNlwxad0KA6a1zuV79F6WdOEX0sXl1LuUHtMmyFAzYN6rMF/KsYPNmji+Jl8i0ii2gIwCOWSVu6jIYMmcK9eXMuI5PEiLmfM+I8o9I4pSK4y9mJNi6DBVjQz1OW/IOWs8uOzvLLm0IZ2pglVMq+76z22A9IixaDi+JG5tKCQUTbywnE/zHKuAJXGvrfbSekUZASxieqdTWQwXMXafjNNx/yY1PvVcrZjoDaV4W6RDbQpNJTrHAWHTll7aMCTFVPLyUuirFtOA+cROhhf5Aa6J8TEWMcmKhXN1CKKwVrBLEvhPIsVUet/UXqwKJr1zliw3mN7BklRbxBT3Uo6oF8QWxfQVYY5JbyqpY9VGUeyhF4hFTGxH3MTfmkPhGGsF+dfUiGm1lSo3LhVK4rh+hr6PQ2+jEZM8NJZ/WxduSQL+6WoOmE4dtGJa9fnjDeWmojxmGaD3GNqNlSXyCc8tAquH4Zw15fOnqznguUGcu0+u/yqxHonwsvRwNJWVfD6tTlqsxdK1leYcn67mgGfPn1pXaWzqfIlu1kKHerfHTSxZsTyMtG5Uj7I1MS+TFaIUmFzAUnXXY+GRwx9QOkJiJZf6VsEvpGvXViKlLnUJWr/mBemOrW6WeVNOgHMhTQqYecIRZS9Xmy/9JQ2mzxqHOrkO6j2Ov2FYirvSPdZUPtLyLX0g60ZxMVL3yc8Re0hpYDeULVMj3LXCL/eNmPa+weQwTDmgTP407iwOPZbewxnPMZ5fEy1xQV1ZmvtynMxT5tg//qdpXl6xnZbAHDuEfewx7csRlcZxQFh47c3NY/wWvvs84L3hZw97su5slBbBO72j4UYvvclfczh1dLAf2mr4hl+mOBW0eO0vLS5wIf1rGrzu5+GrPadyHMdxHMdxHOcL+QE6N+gMZ2UFYgAAAABJRU5ErkJggg==" style="width: 73.5px; height: 18.5px;" width="73.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.5px 8px; transform-origin: 6.5px 8px; unicode-bidi: normal; ">If <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHgAAAAlCAYAAACJdC37AAADhklEQVR4nO1af3HzMAx9HMIgBEJgCIqgDMogDEqhGAohHEKhGEqh3x/OuyhebEtOtvTr9O58t900xdaPJ8kJ4HA4HA6Hw/GZ6ACcdtJ12UmPYyd0AAYAzU76Tjvr+0h0AL6i1Wbk2xV5rtJzRpSdwQy/QpehZ6XeP4sOwUhPAK9p3TPyLYIDRiE/THpSaCb9uSBgNsp9aKm8B3BTyv5ZnDEb9ol8FgPB+C8ER5cwQO+Am9iHNisbAA/sV9s/EtLBL5Tp8aKU+5rkSgFDDJhZwQKykFN1AjJzmJk5Y92hc5wleyGe3xv+B5jLgHfWCTwQnCZra65mPlGm5w62WspsLz07hRvCORwR6IgLllSdarYoX8qyK2y1tBfPrgH3nmv4imhWFHB00NaZdwPraYeZ6mjotTNRvpRlI3RNmJSvqb+EDFQ1TgiRJVt4KmgxNwXsPi3R0yA9U1rW1sC6Y0ltzLxUlt4RzlqCxVmNeKa0b4/Z7iOCP3KM8IJxZGJ2PrCM6m56cI8ltVwNumXN2bK2NhaxUVqh+4HvBi3NyoCexgmOXTHFklG0eqoZQNJHi2W2SkedDTp5dbd1bZn/uPd43+yS4yYpJZ/Sq3UMWUMyQ4NgdwsrDqhotGL6GLE84EX8/X+bw2jYmOZlRsls7RPyMawOJkPyWe30s7X8sGSaIA+7NtfdsIEaDsaIdMTHZYnymsbJ4mBZEi6YS2JNslQ5WF4CrBmDTYB1OD8aZKZU3yCZqRfymnNaHCxHMzJErYOtnTuAZSTH9a4Tf7POX0d30TRsqobLkekB25zJrNQ4WNZ7OaLVNI9mJpUOXOscGeWasSHG0V00mSmXKZK9RtjOqTU2ncp3u6/odwusk8yCptYinZxf87rq6C7act3IZTmnpqOV+tm4yqCyXFlqO/xvm0xlqIw2Kr1j41XZL4GG1US7vMyxGI/JkctCmUAsNSnWZANW0mUqWbnIlc1BM8mY6OFAMEs02R+fUwvW4VxQMHhiJpFBdUJweokNyGhqyPFozRAy+kaUb3feBTJDLHNqzRhYMjr3EScGL5PInqVrYE0wfcO5sMEGwamDVfGBkGfi6lGmNb5psiL3wl/2IGvO6xDsqyl7PSrGI8c+uOJnL4GY7TXvkB07gHfKP8VyZCHHgeCHcXtPGP5F5RuhQfkTWwvOcOe+HXg9uwf8M1mHw+FwOByOd8A/9F+rEvIj56cAAAAASUVORK5CYII=" style="width: 60px; height: 18.5px;" width="60" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 95.5px 8px; transform-origin: 95.5px 8px; unicode-bidi: normal; ">, and given an integer limit x<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 173.5px 8px; transform-origin: 173.5px 8px; unicode-bidi: normal; ">, write a function that returns a sorted array of all <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.5px 8px; transform-origin: 24.5px 8px; unicode-bidi: normal; ">unique<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 36.5px 8px; transform-origin: 36.5px 8px; unicode-bidi: normal; "> values of n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> that are less than or equal to x<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.5px 8px; transform-origin: 27.5px 8px; unicode-bidi: normal; ">-----------</div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27px 8px; transform-origin: 27px 8px; unicode-bidi: normal; ">HINT: <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.5px 8px; transform-origin: 16.5px 8px; unicode-bidi: normal; ">Both R<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and K<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109px 8px; transform-origin: 109px 8px; unicode-bidi: normal; ">, are rational functions and expect <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.5px 8px; transform-origin: 16.5px 8px; unicode-bidi: normal; ">exact<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 169px 8px; transform-origin: 169px 8px; unicode-bidi: normal; "> rational fraction outputs. Therefore, please preserve numerators and denominators for R<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and K<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 233.5px 8px; transform-origin: 233.5px 8px; unicode-bidi: normal; ">, and evaluate decimal expansions only when calculating the output of the function N<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">. </div></div></div>

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Problem 53735. Easy Sequences 58: Curious Prime-Rational Functions

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Problem 53735. Easy Sequences 58: Curious Prime-Rational Functions

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