Additive function: Difference between revisions

Revision as of 18:38, 30 June 2010

Revision as of 01:01, 10 November 2010

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* The '''multiplicity''' of a prime factor ''p'' in ''n'', that is the largest exponent ''m'' for which ''p^m'' divides ''n''.

* ''a''₀(''n'') - the sum of primes dividing ''n'' counting multiplicity, sometimes called sopfr(''n''), the potency of ''n'' or the integer logarithm of ''n'' {{OEIS|A001414}}. For example:

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::''a''₀(2,003) = 2003

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

::''a''₀(54,032,858,972,302) = 1780417

::''a''₀(20,802,650,704,327,415) = 1240681

* The function Ω(''n''), defined as the total number of [[prime number|prime]] factors of ''n'', counting multiple factors multiple times, sometimes called the "[[Big O notation|Big Omega function]]" {{OEIS|A001222}}. For example;

::Ω(1) = 0, since 1 has no prime factors

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::Ω(2,003) = 1

::Ω(54,032,858,972,279) = 3

::Ω(54,032,858,972,302) = 6

::Ω(20,802,650,704,327,415) = 7

Example of arithmetic functions which are additive but not completely additive are:

* ω(''n''), defined as the total number of ''different'' [[prime number|prime]] factors of ''n'' {{OEIS|A001221}}. For example:

::ω(4) = 1

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::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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::''a''₁(2,002) = 33

::''a''₁(2,003) = 2003

::''a''₁(54,032,858,972,279) = 1238665

::''a''₁(54,032,858,972,302) = 1780410

::''a''₁(20,802,650,704,327,415) = 1238677

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== Further reading ==

{{refbegin}}

* Janko Bračič, ''Kolobar aritmetičnih funkcij'' (''[[Ring (algebra)|Ring]] of arithmetical functions''), (Obzornik mat, fiz. '''49''' (2002) 4, pp. 97–108) <fontspan colorstyle="color:darkblue;"> (MSC (2000) 11A25) </fontspan>

{{refend}}