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.

Solve

Solution Stats

2 Solutions
1 Solvers

Last Solution submitted on Jan 05, 2022

Last 200 Solutions

Problem Recent Solvers1

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 52634. Easy Sequences 14: Consecutive Composites

Solution Stats

Last 200 Solutions

Problem Recent Solvers1

Suggested Problems

More from this Author65

Problem Tags

Community Treasure Hunt

Problem 52634. Easy Sequences 14: Consecutive Composites

Solution Stats

Last 200 Solutions

Problem Recent Solvers1

Suggested Problems

More from this Author65

Problem Tags

Community Treasure Hunt

Players