Additive function: Difference between revisions

Revision as of 16:36, 20 September 2006

Revision as of 16:37, 20 September 2006

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Arithmetic functions which are completely additive are:

* The restriction of the [[logarithm|logarithmic function]] to '''N''', ''a''₀(''n'') - the sum of primes dividing ''n'', sometimes called sopfr(''n''). We have ''a''₀(20) = ''a''₀(2² · 5) = 2 + 2+ 5 = 9. Some values: ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001414 OEISSIDN A001414]).

::''a''₀(4) = 4

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:: ...

* ''a''₁(''n'') - the sum of the distinct primes dividing ''n'', sometimes called sopf(''n''). We have ''a''₁(1) = 0, ''a''₁(20) = 2 + 5 = 7. Some more values: ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A008472 OEISSIDN A008472])

::''a''₁(4) = 2

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:: ...

* The function OΩ(''n''), defined as the total number of [[prime number|prime]] factors of ''n'', counting multiple factors multiple times. It is often called "[[Big Omega function]]".This implies OΩ(1) = 0 since 1 has no prime factors. Some more values: ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001222 OEISSIDN A001222])

::OΩ(4) = 2

::OΩ(27) = 3

::OΩ(144) = OΩ(2⁴ · 3²) = OΩ(2⁴) + OΩ(3²) = 4 + 2 = 6

::OΩ(2,000) = OΩ(2⁴ · 5³) = OΩ(2⁴) + OΩ(5³) = 4 + 3 = 7

::OΩ(2,001) = 3

::OΩ(2,002) = 4

::OΩ(2,003) = 1

::OΩ(54,032,858,972,279) = 3

::OΩ(54,032,858,972,302) = 6

::OΩ(20,802,650,704,327,415) = 7

:: ...

* An example of an arithmetic function which is additive but not completely additive is ?ω(''n''), defined as the total number of ''different'' [[prime number|prime]] factors of ''n''. Some values (compare with OΩ(''n'')) ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001221 OEISSIDN A001221])

:

::?ω(4) = 1

::?ω(27) = 1

::?ω(144) = ?ω(2⁴ · 3²) = ?ω(2⁴) + ?ω(3²) = 1 + 1 = 2

::?ω(2,000) = ?ω(2⁴ · 5³) = ?ω(2⁴) + ?ω(5³) = 1 + 1 = 2

::?ω(2,001) = 3

::?ω(2,002) = 4

::?ω(2,003) = 1

::?ω(54,032,858,972,279) = 3

::?ω(54,032,858,972,302) = 5

::?ω(20,802,650,704,327,415) = 5

:: ...

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== References ==

# Janko BracicBračič, ''Kolobar aritmeticniharitmetičnih funkcij'' (''[[Ring (algebra)|Ring]] of arithmetical functions''), (Obzornik mat, fiz. '''49''' (2002) 4, pp 97 - 108) <font color=darkblue> (MSC (2000) 11A25) </font>

== See also ==

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[[it:Funzione additiva]]

[[sv:Additiv funktion]]

[[zh:????加性函數]]