Arithmetic function: Difference between revisions

Revision as of 10:49, 30 September 2002

Revision as of 13:57, 30 September 2002

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:hence ''c''₄(1)=4 ≠ 1.

* the [[Partition function]] ''P''(''n''), the [[Partition function]] - the number of representations of ''n'' as a sum of positive integers, where we don't distinguish between different orders of the summands. For instance: ''P''(2 · 5) = ''P''(10) = 42 and ''P''(2)''P''(5) = 2 · 7 = 14 ≠ 42.

* π (''n''), the [[Prime number theorem|Prime counting function]] π (''n'') - the number of [[prime number|primes]] less than or equal to a given number ''n''. We have π(1) = 0 ≠ 1, π (2 · 5) = π(10) = 4 and π(2) π(5) = 1 · 3 = 3 ≠ 4.

* ''a''_''0''(''n'') - the sum of primes dividing ''n'', sometimes called sopfr(''n''). We have ''a''_''0''(1) = 0 ≠ 1, ''a''_''0''(2 · 5) = ''a''_''0''(10) = 7 and ''a''_''0''(2) ''a''_''0''(5) = 2 · 5 = 10 ≠ 7([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001414 SIDN A001414]).

* ''a''_''1''(''n'') - the sum of the distinct primes dividing ''n'', sometimes called sopf(''n''). We have ''a''_''1''(1) = 0 ≠ 1, ''a''_''1''(2 · 5) = ''a''_''1''(10) = 7 and ''a''_''1''(2) ''a''_''1''(5) = 2 · 5 = 10 ≠ 7([http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A008472 SIDN A008472]).