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tat was false for 54,032,858,972,279 and i gave the detail of the computing |
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* The restriction of the [[Logarithm|logarithmic function]] to <math>\N.</math>
* The '''multiplicity''' of a [[Prime number|prime]] factor ''p'' in ''n'', that is the largest exponent ''m'' for which ''p<sup>m</sup>'' [[Divisor|divides]] ''n''.
* {{anchor|Integer logarithm}} ''a''<sub>0</sub>(''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:
::''a''<sub>0</sub>(4) = 2 + 2 = 4
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::Ω(2002) = 4
::Ω(2003) = 1
::Ω(54,032,858,972,279) = Ω(11 ⋅ 1993<sup>2</sup> ⋅ 1236661) = 4
::Ω(54,032,858,972,302) = Ω(2 ⋅ 7<sup>2</sup> ⋅ 149 ⋅ 2081 ⋅ 1778171) = 6
::Ω(20,802,650,704,327,415) = Ω(5 ⋅ 7 ⋅ 11<sup>2</sup> ⋅ 1993<sup>2</sup> ⋅ 1236661) = 7.
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From any additive function <math>f(n)</math> it is possible to create a related {{em|[[multiplicative function]]}} <math>g(n),</math> which is a function with the property that whenever <math>a</math> and <math>b</math> are coprime then:
<math display=block>g(a b) = g(a) \times g(b).</math>
One such example is <math>g(n) = 2^{f(n)}.</math> Likewise if <math>f(n)</math> is completely additive, then <math>g(n) = 2^{f(n)} </math> is completely multiplicative. More generally, we could consider the function <math>g(n) = c^{f(n)} </math>, where <math>c</math> is a nonzero real constant.
== Summatory functions ==
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