Additive function: Difference between revisions

Revision as of 18:24, 24 February 2006

Revision as of 10:14, 15 July 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 SIDN A001414]).

::''a''₀(4) = 4

::''a''₀(27) = 9

::''a''₀(144) = ''a''₀(2⁴ ·· 3²) = ''a''₀(2⁴) + ''a''₀(3²) = 8 + 6 = 14

::''a''₀(2,000) = ''a''₀(2⁴ ·· 5³) = ''a''₀(2⁴) + ''a''₀(5³) = 8 + 15 = 23

::''a''₀(2,003) = 2003

::''a''₀(54,032,858,972,279) = 1240658

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::''a''₁(4) = 2

::''a''₁(27) = 3

::''a''₁(144) = ''a''₁(2⁴ ·· 3²) = ''a''₁(2⁴) + ''a''₁(3²) = 2 + 3 = 5

::''a''₁(2,000) = ''a''₁(2⁴ ·· 5³) = ''a''₁(2⁴) + ''a''₁(5³) = 2 + 5 = 7

::''a''₁(2,001) = 55

::''a''₁(2,002) = 33

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:: ...

* The function ΩΩ(''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 ΩΩ(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 SIDN A001222])

::ΩΩ(4) = 2

::ΩΩ(27) = 3

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

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

::ΩΩ(2,001) = 3

::ΩΩ(2,002) = 4

::ΩΩ(2,003) = 1

::ΩΩ(54,032,858,972,279) = 3

::ΩΩ(54,032,858,972,302) = 6

::ΩΩ(20,802,650,704,327,415) = 7

:: ...

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

:

::ωω(4) = 1

::ωω(27) = 1

::ωω(144) = ωω(2⁴ ·· 3²) = ωω(2⁴) + ωω(3²) = 1 + 1 = 2

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

::ωω(2,001) = 3

::ωω(2,002) = 4

::ωω(2,003) = 1

::ωω(54,032,858,972,279) = 3

::ωω(54,032,858,972,302) = 5

::ωω(20,802,650,704,327,415) = 5

:: ...

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== References ==

# Janko Bra&#269;i&#269;Bračič, ''Kolobar aritmeti&#269;niharitmetičnih 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]]

[[zh:&#21152;&#24615;&#20989;&#25976;]]