Problem 52634. Easy Sequences 14: Consecutive Composites

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>Generating a given number of consecutive numbers all of which are composites is easy. For example the following is a sequence of 100 consecutive composite 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>101!+2, 100!+3, 101!+4, ... 101!+101</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>These numbers are huge. The first number of the sequence is around 9.4259 × 10 ^ 159. A smaller solution can be found by using \"primorials\" instead of factorials. Primorial is the product of all primes less than or equal to a given number. If we use primorials the first number is reduced to:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[   >> prod(primes(101)) + 2\n   >> ans =\n   2.3286e+38]]></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>Although, this is a significant size reduction, the smallest possible 100 consecutive composites series, is still way less than that.  The smallest 100 consecutive composites series,starts with </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>370262, </w:t></w:r><w:r><w:t>that is:</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>370262, 370263, 370264, ... , 370361</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>if we define the set 'C' as the smallest consecutive 'n' composite numbers, the function 'f(n)' as the smallest element in 'C', and 'F' is the set that contains all f(x)'s for x = 1,2,3...n, </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>write the function S(n), which is the sum of all elements of F</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: 20.440000534057617px; 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: normal; text-decoration: none; white-space: normal; "><div style="block-size: 347px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; 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; text-align: left; 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; "><span style="">Generating a given number of consecutive numbers all of which are composites is easy. For example the following is a sequence of 100 consecutive composite numbers:</span></span></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; text-align: left; 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; "><span style="">           </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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width="255.5" height="18" style="width: 255.5px; height: 18px;"></span></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; text-align: left; 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; "><span style="">These numbers are huge. The first number of the sequence is around 9.4259 × 10 ^ 159. A smaller solution can be found by using "primorials" instead of factorials. Primorial is the product of all primes less than or equal to a given number. If we use primorials the first number is reduced to:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; 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; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; tab-size: 4; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">   &gt;&gt; prod(primes(101)) + 2</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; 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; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; tab-size: 4; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">   &gt;&gt; ans =</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; 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; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; tab-size: 4; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">   2.3286e+38</span></span></div></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; 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; "><span style="">Although, this is a significant size reduction, the smallest possible 100 consecutive composites series, is still way less than that.  The smallest 100 consecutive composites series,starts with </span></span><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; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; ">370262, </span></span><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; "><span style="">that is:</span></span></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; text-align: left; 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; "><span style="">            </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="219.5" height="18" style="width: 219.5px; height: 18px;"></span></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; text-align: left; 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; "><span style="">if we define the set 'C' as the smallest consecutive 'n' composite numbers, the function 'f(n)' as the smallest element in 'C', and 'F' is the set that contains all f(x)'s for x = 1,2,3...n, </span></span><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; "><span style="font-weight: bold; ">write the function S(n), which is the sum of all elements of F</span></span><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; "><span style="">.</span></span></div></div></div>

Generating a given number of consecutive numbers all of which are composites is easy. For example the following is a sequence of 100 consecutive composite numbers:
           
These numbers are huge. The first number of the sequence is around 9.4259 × 10 ^ 159. A smaller solution can be found by using "primorials" instead of factorials. Primorial is the product of all primes less than or equal to a given number. If we use primorials the first number is reduced to:
   >> prod(primes(101)) + 2
   >> ans =
   2.3286e+38
Although, this is a significant size reduction, the smallest possible 100 consecutive composites series, is still way less than that.  The smallest 100 consecutive composites series,starts with 370262, that is:
            
if we define the set 'C' as the smallest consecutive 'n' composite numbers, the function 'f(n)' as the smallest element in 'C', and 'F' is the set that contains all f(x)'s for x = 1,2,3...n, write the function S(n), which is the sum of all elements of F.

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Brandon on 1 Jun 2023

Prime gaps.
You're looking for prime gaps.

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Problem 52634. Easy Sequences 14: Consecutive Composites

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Problem 52634. Easy Sequences 14: Consecutive Composites

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