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Lcm of all numbers so far - Revision history
2026-06-05T01:03:54Z
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Vipul: /* Behavior */
2012-07-03T17:55:45Z
<p><span dir="auto"><span class="autocomment">Behavior</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:55, 3 July 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Behavior==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Behavior==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{oeis|A003418}}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The <ins style="font-weight: bold; text-decoration: none;">initial </ins>values <ins style="font-weight: bold; text-decoration: none;">of the lcm of all numbers so far </ins>for <math><ins style="font-weight: bold; text-decoration: none;">k</ins>=0,1,2,3,4,<ins style="font-weight: bold; text-decoration: none;">\dots</ins></math> are <ins style="font-weight: bold; text-decoration: none;">given as: </ins><<ins style="font-weight: bold; text-decoration: none;">section begin="list"/</ins>><ins style="font-weight: bold; text-decoration: none;">[[</ins>1<ins style="font-weight: bold; text-decoration: none;">]]</ins>, <ins style="font-weight: bold; text-decoration: none;">[[</ins>1<ins style="font-weight: bold; text-decoration: none;">]]</ins>, <ins style="font-weight: bold; text-decoration: none;">[[</ins>2<ins style="font-weight: bold; text-decoration: none;">]]</ins>, <ins style="font-weight: bold; text-decoration: none;">[[</ins>6<ins style="font-weight: bold; text-decoration: none;">]]</ins>, <ins style="font-weight: bold; text-decoration: none;">[[</ins>12<ins style="font-weight: bold; text-decoration: none;">]]</ins>, <ins style="font-weight: bold; text-decoration: none;">[[</ins>60<ins style="font-weight: bold; text-decoration: none;">]]</ins>, <ins style="font-weight: bold; text-decoration: none;">[[</ins>60<ins style="font-weight: bold; text-decoration: none;">]]</ins>, <ins style="font-weight: bold; text-decoration: none;">[[</ins>420<ins style="font-weight: bold; text-decoration: none;">]], <toggledisplay>[[840]], [[2520]]</ins></<ins style="font-weight: bold; text-decoration: none;">toggledisplay> [[Oeis:A003418|View list on OEIS]]<section end="list"/</ins>></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">===Initial values===</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The values for <math><del style="font-weight: bold; text-decoration: none;">n </del>= 0,1,2,3,4,<del style="font-weight: bold; text-decoration: none;">5,6,7</del></math> are <<del style="font-weight: bold; text-decoration: none;">math</del>>1,1,2,6,12,60,60,420</<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Growth===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Growth===</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is the exponential of the [[second Chebyshev function]]. More details on the growth are to be found in the page on the second Chebyshev function.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is the exponential of the [[second Chebyshev function]]. More details on the growth are to be found in the page on the second Chebyshev function.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Relation with other functions==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Relation with other functions==</div></td></tr>
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Vipul
https://number.subwiki.org/w/index.php?title=Lcm_of_all_numbers_so_far&diff=941&oldid=prev
Vipul at 17:53, 3 July 2012
2012-07-03T17:53:20Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:53, 3 July 2012</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Let <math>n</math> be a natural number. The '''lcm of all numbers so far''' for <math>n<math> is defined as:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Let <math>n</math> be a natural number. The '''lcm of all numbers so far''' for <math>n<<ins style="font-weight: bold; text-decoration: none;">/</ins>math> is defined as:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* The least common multiple of all numbers from <math>1</math> to <math>n</math>, i.e., as:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* The least common multiple of all numbers from <math>1</math> to <math>n</math>, i.e., as:</div></td></tr>
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Vipul
https://number.subwiki.org/w/index.php?title=Lcm_of_all_numbers_so_far&diff=265&oldid=prev
Vipul: Created page with '==Definition== Let <math>n</math> be a natural number. The '''lcm of all numbers so far''' for <math>n<math> is defined as: * The least common multiple of all numbers from <mat...'
2009-04-30T20:29:25Z
<p>Created page with '==Definition== Let <math>n</math> be a natural number. The '''lcm of all numbers so far''' for <math>n<math> is defined as: * The least common multiple of all numbers from <mat...'</p>
<p><b>New page</b></p><div>==Definition==<br />
<br />
Let <math>n</math> be a natural number. The '''lcm of all numbers so far''' for <math>n<math> is defined as:<br />
<br />
* The least common multiple of all numbers from <math>1</math> to <math>n</math>, i.e., as:<br />
<br />
<math>\operatorname{lcm}\{1,2,\dots,n\}</math>.<br />
<br />
* The product of all primes powers <math>p^k</math> for which <math>p^k \le n</math> but <math>p^{k+1} > n</math>.<br />
* The [[groupprops:exponent of a group|exponent]] of the [[groupprops:symmetric group|symmetric group]] of degree <math>n</math>.<br />
* It is the exponential of the [[defining ingredient::second Chebyshev function]].<br />
==Behavior==<br />
<br />
{{oeis|A003418}}<br />
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===Initial values===<br />
<br />
The values for <math>n = 0,1,2,3,4,5,6,7</math> are <math>1,1,2,6,12,60,60,420</math>.<br />
<br />
===Growth===<br />
<br />
The lcm of all numbers so far has approximately exponential growth in <math>n</math>. Moreover, it is not strictly increasing as a function of <math>n</math>, and it increases in value ''only'' at prime powers. At the prime power <math>p^k</math>, it gets multiplied by <math>p</math>.<br />
<br />
It is the exponential of the [[second Chebyshev function]]. More details on the growth are to be found in the page on the second Chebyshev function.<br />
==Relation with other functions==<br />
<br />
* [[Factorial]]<br />
* [[Primorial]], [[Primorial of the second kind]]<br />
<br />
===Logarithm===<br />
<br />
The logarithm of the lcm of all numbers so far is equal to the [[second Chebyshev function]].</div>
Vipul