Problem 58249. Easy Sequences 114: Counting Unique Numbers

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:rPr><w:b/></w:rPr><w:t>A </w:t></w:r><w:hyperlink w:docLocation=\"https://www.javatpoint.com/unique-number-in-java-program\"><w:r><w:rPr><w:b/></w:rPr><w:t>Unique Number</w:t></w:r></w:hyperlink><w:r><w:rPr><w:b/></w:rPr><w:t> is defined as a positive integer with no repeated digits.</w:t></w:r><w:r><w:t> This means that </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>123</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>1234</w:t></w:r></w:customXml><w:r><w:t> are unique numbers, but </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>114</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>1144</w:t></w:r></w:customXml><w:r><w:t> are not.</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>Given the number </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>B</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, write a program the calculates the number of unique numbers in base-B. </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: </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:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\cdot</w:t></w:r></w:customXml><w:r><w:t> In base-2, there are only </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>2</w:t></w:r><w:r><w:t> unique 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>[1,\\ 10]</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> </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>\\cdot</w:t></w:r></w:customXml><w:r><w:t> In base-3, there are </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>10</w:t></w:r><w:r><w:t> unique 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>[1,\\ 2,\\ 10,\\ 12,\\ 20,\\ 21,\\ 102,\\ 120,\\ 201,\\ 210]</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> </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>\\cdot</w:t></w:r></w:customXml><w:r><w:t> In base-4, there are </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>48</w:t></w:r><w:r><w:t> unique numbers: </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:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>[1,\\ 2,\\ 3,\\ 10,\\ 12,\\ 13,\\ 20,\\ 21,\\ 23,\\ 30,\\ 31,\\ 32,\\ 102,\\ 103,\\ 120,\\ 123,\\ 130,\\ 132,\\ 201,\\ 203,\\\\ \\ 210,\\ 213,\\ 230,\\ 231,\\ 301 ,\\ 302,\\ 310,\\ 312,\\ 320,\\ 321,\\ 1023,\\ 1032,\\ 1203,\\ 1230,\\ 1302,\\\\ \\ 1320,\\ 2013,\\ 2031,\\ 2103,\\ 2130,\\ 2301,\\ 2310,\\ 3012,\\ 3021,\\ 3102,\\ 3120,\\ 3201,\\ 3210].</w:t></w:r></w:customXml></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:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\cdot</w:t></w:r></w:customXml><w:r><w:t> And in base-10, there are </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>8,877,690</w:t></w:r><w:r><w:t> unique numbers.</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>The output could be greater than </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>flintmax</w:t></w:r><w:r><w:t>, therefore </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>please present your output as character strings</w:t></w:r><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: 322px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 161px; transform-origin: 407px 161px; vertical-align: baseline; "><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; ">A <a target='_blank' href = "https://www.javatpoint.com/unique-number-in-java-program">Unique Number</a><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: 196px 8px; transform-origin: 196px 8px; unicode-bidi: normal; "> is defined as a positive integer with no repeated digits.<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: 54.5px 8px; transform-origin: 54.5px 8px; unicode-bidi: normal; "> This means that <img src="data:image/png;base64,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" style="width: 25.5px; height: 18px;" width="25.5" 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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" style="width: 33px; height: 18px;" width="33" 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: 14.5px 8px; transform-origin: 14.5px 8px; unicode-bidi: normal; "> are unique numbers, but <img src="data:image/png;base64,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" style="width: 25.5px; height: 18px;" width="25.5" 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: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" style="width: 33px; height: 18px;" width="33" 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: 26.5px 8px; transform-origin: 26.5px 8px; unicode-bidi: normal; "> are not.</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: 65.5px 8px; transform-origin: 65.5px 8px; unicode-bidi: normal; ">Given the number B<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: 260px 8px; transform-origin: 260px 8px; unicode-bidi: normal; ">, write a program the calculates the number of unique numbers in base-B. </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: 43px 8px; transform-origin: 43px 8px; unicode-bidi: normal; ">For example: </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: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; "> ⋅<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: 81.5px 8px; transform-origin: 81.5px 8px; unicode-bidi: normal; "> In base-2, there are only <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; ">2<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: 57px 8px; transform-origin: 57px 8px; unicode-bidi: normal; "> unique numbers: <img src="data:image/png;base64,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" style="width: 47px; height: 18.5px;" width="47" 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; ">. </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: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; "> ⋅<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: 66.5px 8px; transform-origin: 66.5px 8px; unicode-bidi: normal; "> In base-3, there are <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; ">10<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: 59px 8px; transform-origin: 59px 8px; unicode-bidi: normal; "> unique numbers: <img src="data:image/png;base64,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" style="width: 275.5px; height: 18.5px;" width="275.5" 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: 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: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; "> ⋅<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: 66.5px 8px; transform-origin: 66.5px 8px; unicode-bidi: normal; "> In base-4, there are <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; ">48<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: 95px 8px; transform-origin: 95px 8px; unicode-bidi: normal; "> unique numbers: </div><div style="block-size: 60.5px; 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 30.25px; text-align: left; transform-origin: 384px 30.25px; 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: 40px 8px; transform-origin: 40px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 554px; height: 60.5px;" width="554" height="60.5"></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: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; "> ⋅<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: 85px 8px; transform-origin: 85px 8px; unicode-bidi: normal; "> And in base-10, there are <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: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; ">8,877,690<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: 55px 8px; transform-origin: 55px 8px; unicode-bidi: normal; "> unique numbers.</div><div style="block-size: 21.5px; 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 10.75px; text-align: left; transform-origin: 384px 10.75px; 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: 106.5px 8px; transform-origin: 106.5px 8px; unicode-bidi: normal; ">The output could be greater than <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: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; ">flintmax<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: 35px 8px; transform-origin: 35px 8px; unicode-bidi: normal; ">, therefore <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: 167.5px 8px; transform-origin: 167.5px 8px; unicode-bidi: normal; ">please present your output as character strings<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></div>

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Problem 58249. Easy Sequences 114: Counting Unique Numbers

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Problem 58249. Easy Sequences 114: Counting Unique Numbers

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