Arithmetic function: Difference between revisions

Revision as of 09:16, 29 October 2006

Revision as of 14:44, 6 November 2006

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The articles on additive and multiplicative functions contain several examples of arithmetic functions. Here are some examples that are neither additive nor multiplicative:

* ''cr''₄(''n'') - the number of ways that ''n'' can be expressed as the sum of four squares of nonnegative integers, where we distinguish between different orders of the summands. For example:

::1 = 1²+0²+0²+0² = 0²+1²+0²+0² = 0²+0²+1²+0² = 0²+0²+0²+1²,

:hence ''cr''₄(1)=4.

* ''P''(''n''), the [[Partition function (number theory)|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.