Revision as of 08:27, 2 February 2007 edit David Eppstein (talk \| contribs) Autopatrolled, Administrators 235,669 edits m →Examples: two different examples shared a bullet; separate them ← Previous edit		Revision as of 16:04, 11 February 2007 edit undo Fudo (talk \| contribs) 240 edits →Examples: a_1 is not completely additive! Next edit →
Line 32: ::''a''<sub>0</sub>(54,032,858,972,302) = 1780417 ::''a''<sub>0</sub>(20,802,650,704,327,415) = 1240681 :: ...▼ * ''a''<sub>1</sub>(''n'') - the sum of the distinct primes dividing ''n'', sometimes called sopf(''n''). We have ''a''<sub>1</sub>(1) = 0, ''a''<sub>1</sub>(20) = 2 + 5 = 7. Some more values: ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A008472 OEIS A008472]) ::''a''<sub>1</sub>(4) = 2▼ ::''a''<sub>1</sub>(27) = 3▼ ::''a''<sub>1</sub>(144) = ''a''<sub>1</sub>(2<sup>4</sup> · 3<sup>2</sup>) = ''a''<sub>1</sub>(2<sup>4</sup>) + ''a''<sub>1</sub>(3<sup>2</sup>) = 2 + 3 = 5▼ ::''a''<sub>1</sub>(2,000) = ''a''<sub>1</sub>(2<sup>4</sup> · 5<sup>3</sup>) = ''a''<sub>1</sub>(2<sup>4</sup>) + ''a''<sub>1</sub>(5<sup>3</sup>) = 2 + 5 = 7▼ ::''a''<sub>1</sub>(2,001) = 55▼ ::''a''<sub>1</sub>(2,002) = 33▼ ::''a''<sub>1</sub>(2,003) = 2003▼ ::''a''<sub>1</sub>(54,032,858,972,279) = 1238665 ▼ ::''a''<sub>1</sub>(54,032,858,972,302) = 1780410 ▼ ::''a''<sub>1</sub>(20,802,650,704,327,415) = 1238677▼ :: ... Line 62 ⟶ 48: :: ... * AnThe ~~example~~function of''a''<sub>1</sub>(''n'') an- ~~arithmetic~~the ~~function~~sum ~~which~~of isthe ~~additive~~distinct ~~but~~primes ~~not~~dividing ~~completely~~''n'', ~~additive~~sometimes iscalled ωsopf(''n''), ~~defined~~is asadditive ~~the~~but ~~total~~not ~~number~~completely ofadditive. We have ''~~different~~a''<sub>1</sub>(1) ~~[[prime~~= ~~number\|prime]] factors of~~0, ''na''.<sub>1</sub>(20) ~~Some~~= ~~values~~2 ~~(compare~~+ ~~with~~5 ~~Ω(''n''))~~= 7. Some more values: ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=~~A001221~~A008472 OEIS ~~A001221~~A008472]) ▲::''a''<sub>1</sub>(4) = 2 ▲::''a''<sub>1</sub>(27) = 3 ▲::''a''<sub>1</sub>(144) = ''a''<sub>1</sub>(2<sup>4</sup> · 3<sup>2</sup>) = ''a''<sub>1</sub>(2<sup>4</sup>) + ''a''<sub>1</sub>(3<sup>2</sup>) = 2 + 3 = 5 ▲::''a''<sub>1</sub>(2,000) = ''a''<sub>1</sub>(2<sup>4</sup> · 5<sup>3</sup>) = ''a''<sub>1</sub>(2<sup>4</sup>) + ''a''<sub>1</sub>(5<sup>3</sup>) = 2 + 5 = 7 ▲::''a''<sub>1</sub>(2,001) = 55 ▲::''a''<sub>1</sub>(2,002) = 33 ▲::''a''<sub>1</sub>(2,003) = 2003 ▲::''a''<sub>1</sub>(54,032,858,972,279) = 1238665 ▲::''a''<sub>1</sub>(54,032,858,972,302) = 1780410 ▲::''a''<sub>1</sub>(20,802,650,704,327,415) = 1238677 ▲:: ... * Another 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 Ω(''n'')) ([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001221 OEIS A001221]) :

Additive function: Difference between revisions