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Difference between revisions of "Smarandache function"

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Given a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s1303901.png" />, the value of the Smarandache function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s1303902.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s1303903.png" /> is the smallest natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s1303904.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s1303905.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s1303906.png" />. An elementary observation is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s1303907.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s1303908.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s1303909.png" /> is a prime number or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130390/s13039010.png" />.
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Given a natural number $n$, the value of the Smarandache function $ \eta $ at $n$ is the smallest natural number $m$ such that $n$ divides $m!$. An elementary observation is that $\eta ( n ) \leq n$, and that $\eta ( n ) = n$ if and only if $n$ is a prime number or equal to $4$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Smarandache,  "A function in number theory"  ''Smarandache Function J.'' , '''1'''  (1990)  pp. 3–65</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  F. Smarandache,  "A function in number theory"  ''Smarandache Function J.'' , '''1'''  (1990)  pp. 3–65</td></tr></table>

Latest revision as of 16:55, 1 July 2020

Given a natural number $n$, the value of the Smarandache function $ \eta $ at $n$ is the smallest natural number $m$ such that $n$ divides $m!$. An elementary observation is that $\eta ( n ) \leq n$, and that $\eta ( n ) = n$ if and only if $n$ is a prime number or equal to $4$.

References

[a1] F. Smarandache, "A function in number theory" Smarandache Function J. , 1 (1990) pp. 3–65
How to Cite This Entry:
Smarandache function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smarandache_function&oldid=50070
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article