Problem 52784. Easy Sequences 38: Prime Number Delta

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>The </w:t></w:r><w:hyperlink w:docLocation=\"https://www.britannica.com/science/prime-number-theorem\"><w:r><w:t>Prime Number Theorem </w:t></w:r></w:hyperlink><w:r><w:t>states that:</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>\\lim_{x\\rightarrow \\infty}\\ \\frac{\\pi(x)}{x/log(x)}=1</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>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>\\pi(x)</w:t></w:r></w:customXml><w:r><w:t>is the prime counting function (number of 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>\\leq x</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>log(x)</w:t></w:r></w:customXml><w:r><w:t>is the natural logarithm 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>x</w:t></w:r></w:customXml><w:r><w:t>. The convergence of the abovementioned limit, first conjectured by Legendre in 1798, is now well established.</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>In this exercise we are more concerned with the difference 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>\\Delta</w:t></w:r></w:customXml><w:r><w:t>, defined as follows:</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>\\Delta (x)=\\pi(x)-\\left [ \\frac{x}{log(x)} \\right ]</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>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>[ \\ ] </w:t></w:r></w:customXml><w:r><w:t> means rounding-off to nearest integer. </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>\\Delta</w:t></w:r></w:customXml><w:r><w:t> appears divergent and seems to increase without bound as </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:t> increases. </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 a 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>d</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, our goal is to find the value of integer </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> when </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>\\Delta(x)</w:t></w:r></w:customXml><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/><w:u/></w:rPr><w:t>first exceeds</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><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>d</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>As an example, 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>d=10</w:t></w:r></w:customXml><w:r><w:t>, the value 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>x</w:t></w:r></w:customXml><w:r><w:t> is </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>283</w:t></w:r></w:customXml><w:r><w:t> since:</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>\\Delta (283)=\\pi(283)-\\left [ \\frac{283}{log(283)} \\right ]=61-50=11&gt;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>and for all </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&lt;283</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>\\Delta \\le10</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: 377px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 188.5px; transform-origin: 407px 188.5px; vertical-align: baseline; "><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: 14.5px 8px; transform-origin: 14.5px 8px; unicode-bidi: normal; ">The <a target='_blank' href = "https://www.britannica.com/science/prime-number-theorem">Prime Number Theorem </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: 35px 8px; transform-origin: 35px 8px; unicode-bidi: normal; ">states that:</div><div style="block-size: 36px; 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 18px; text-align: left; transform-origin: 384px 18px; 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: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 115px; height: 36px;" width="115" height="36"></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: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; ">where <img src="data:image/png;base64,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" style="width: 31px; height: 18.5px;" width="31" 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: 180px 8px; transform-origin: 180px 8px; unicode-bidi: normal; ">is the prime counting function (number of prime numbers <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADIAAAAkCAYAAADLsGk3AAABMUlEQVRYhe2XbQ2DMBCGHw91gAEMoAAFc4CDOZgFNCABD7MwDVjofrRkx9c2KC2Q3JNclrSB68t9daAoiqIo58IA5dGHCMEAd6DzdjmkAOvtceiJVrIkIDvyUGu4vIAMqEknIAeKmXXj11f77QVY4gsogYbhx5Id8Cb2Wpyon6QU0JML373Pxq+VOBEP/9tMnp5hLKAmfQ203vcTJ7DbcoZCvMj6l1T8GcqduAv/TwKH7Di9UgoqmKZXMEcJijZglwTFqB8DvPh0qCikmCkNw4YTNfqxpnyFi4askyQ36rGgkNtvzjAC4zox7Fj8S/SCXhueueEi0DG8msh5krFDK16D4f+clvPC4tJKUv3YPw05LlVa5r+0wTWWpf2vFDvYKbCBdpq/um2gRe8qiqIo1+YNTlOaOCB9ziIAAAAASUVORK5CYII=" style="width: 25px; height: 18px;" width="25" 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: 20.5px 8px; transform-origin: 20.5px 8px; unicode-bidi: normal; ">), and <img src="data:image/png;base64,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" style="width: 41.5px; height: 18.5px;" width="41.5" 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: 82px 8px; transform-origin: 82px 8px; unicode-bidi: normal; ">is the natural logarithm of 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: 18.5px 8px; transform-origin: 18.5px 8px; unicode-bidi: normal; ">. The convergence of the abovementioned limit, first conjectured by Legendre in 1798, is now well established.</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: 211.5px 8px; transform-origin: 211.5px 8px; unicode-bidi: normal; ">In this exercise we are more concerned with the difference function Δ<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: 62px 8px; transform-origin: 62px 8px; unicode-bidi: normal; ">, defined as follows:</div><div style="block-size: 35.5px; 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 17.75px; text-align: left; transform-origin: 384px 17.75px; 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: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 149px; height: 35.5px;" width="149" height="35.5"></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: 57px 8px; transform-origin: 57px 8px; unicode-bidi: normal; ">where the symbol <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACkAAAAlCAYAAADfosCNAAAAfElEQVRYhe3YwQ2AIBBE0amFOiiEDujA2uyBovTCgRgkys7xv2Svw49HJUCSdLxcDu7mxfZvl6Qm6XxcCUaWyWbr721FRr/aV1lEehDpQqQLkS5EuhDpQqQLkS5EuhDpQqQLkS6hyNoHxkvBoDTZrJHI2W39sxkci20ALje6019blrOi3wAAAABJRU5ErkJggg==" style="width: 20.5px; height: 18.5px;" width="20.5" 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: 125px 8px; transform-origin: 125px 8px; unicode-bidi: normal; "> means rounding-off to nearest integer. Δ<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: 181px 8px; transform-origin: 181px 8px; unicode-bidi: normal; "> appears divergent and seems to increase without bound as 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: 36.5px 8px; transform-origin: 36.5px 8px; unicode-bidi: normal; "> increases. </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: 58px 8px; transform-origin: 58px 8px; unicode-bidi: normal; ">Given a number d<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; ">, our goal is to find the value of integer 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: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; "> when <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEIAAAAlCAYAAAD2pT8KAAACUUlEQVRoge2YbY3DMAyGHw5lMAIlMARFUAZjUAajMAwHoRxKYRhGYfcj8eqL2jRO0upO11fKj2mq7divP2I4ceLEid+PFugqyrtVlHUYWmAEmooyux1kLqKtpKQFpkqyQvQ7yv7gBdwLZTRezrXcnFUMwGMv4T3w9qfE2yM7GunRAE/q1p8PnsyOyC1KV//9pZZREfQ45lVNEbmAnGemgiPYIJAUrNpJRtzlS1jR+u92oesKHjibq+DCfHFdJyajnDvl9cUKsbetIezBnGtCN3GGJboTdufhdV5Zrist8e4jLCxOD2GDbpkDeax441IsBY3XOSldelBq/O+UgLypUJcGHAN0NMQ5clLmAYnMkKi3UTofSpdQfAK+mJ0Ro74lAKvGvLzCENq4pf9DSNdJdcTSt29czt+xRVgKfTZihUYiHEZqDSWOgJ+paK0zwppsPIlTSufoVoRKHaF1WTtAkSM6totQOGTF2mKpI3SBtrbf3G4FOC+mfKwHrNhjTApsriNuSo/1sZZdLCV6Kb1XD1hb0co1KKxH1tdvzjfAPJamUlAPWDHn5VRveUX2zLluobkEtTfqzaKwzt+YA4Xelhz/Yo7mUp24Ea9jotP82pX3QMlZY4U4ORadDnfhFsdMnUrhPCGbqBhGMtIxfEfknphxW4ZtPfVfG/9rpDh+EbL4rHHWev3WYmZg3lksXbLb+D+Uld02j8Cdwtk/ARf234sWo8FFykxZA0byZ5ZDIW2xysIkwK4b7D0gO4Wazuj5Y04QyPapFo7ch544ceLE/8E3g1EBL/EkZZYAAAAASUVORK5CYII=" style="width: 33px; height: 18.5px;" width="33" 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; "> <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: 46px 8px; transform-origin: 46px 8px; unicode-bidi: normal; ">first exceeds<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; "> d<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: 55.5px 8px; transform-origin: 55.5px 8px; unicode-bidi: normal; ">As an example, if <img src="data:image/png;base64,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" style="width: 45px; height: 18px;" width="45" 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: 43px 8px; transform-origin: 43px 8px; unicode-bidi: normal; ">, the value of 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: 9px 8px; transform-origin: 9px 8px; unicode-bidi: normal; "> is <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: 20.5px 8px; transform-origin: 20.5px 8px; unicode-bidi: normal; "> since:</div><div style="block-size: 35.5px; 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 17.75px; text-align: left; transform-origin: 384px 17.75px; 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: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" style="width: 327px; height: 35.5px;" width="327" height="35.5"><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: 33.5px 8px; transform-origin: 33.5px 8px; unicode-bidi: normal; ">and for all <img src="data:image/png;base64,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" 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Problem 52784. Easy Sequences 38: Prime Number Delta

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Problem 52784. Easy Sequences 38: Prime Number Delta

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