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nonmultiplicative function
In number theory, a nonmultiplicative function is an arithmetic function which is not multiplicative.
Examples
Some examples of a nonmultiplicative functions are the arithmetic functions:

$c_{{4}}(n)$  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:
$1=1^{{2}}+0^{{2}}+0^{{2}}+0^{{2}}=0^{{2}}+1^{{2}}+0^{{2}}+0^{{2}}+0^{{2}}=0^{{% 2}}+0^{{2}}+1^{{2}}+0^{{2}}=0^{{2}}+0^{{2}}+0^{{2}}+1^{{2}}\;,$ hence $c_{{4}}(1)=4\neq 1\;.$

The partition function $P(n)$  the number of ordered representations of $n$ as a sum of positive integers. For instance:
$P(2\cdot 5)=P(10)=42\quad\hbox{and}$ $P(2)P(5)=2\cdot 7=14\neq 42\;.$ 
The prime counting function $\pi(n)$. Here we first have $\pi(1)=0\neq 1$ and then we have as yet for example:
$\pi(2\cdot 5)=\pi(10)=4\quad\hbox{and}$ $\pi(2)\pi(5)=1\cdot 3=3\neq 4\;.$ 
The Mangoldt function $\Lambda(n)$. $\Lambda(1)=\ln 1\neq 1$ and for example:
$\Lambda(2\cdot 5)=\Lambda(10)=0\quad\hbox{and}$ $\Lambda(2)\Lambda(5)=\ln 2\cdot\ln 5\neq 0\;.$ We would think that for some $n$ multiplicativity of $\Lambda(n)$ would be true as in:
$\Lambda(2\cdot 6)=\Lambda(12)=0\quad\hbox{and}$ $\Lambda(2)\Lambda(6)=\ln 2\cdot 0=0\;,$ but we have to write:
$\Lambda(2^{{2}})\Lambda(3)=\ln 2\cdot\ln 3\neq 0\;.$
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Corrections
reclassify by rmilson ✓
grammar fixups by djao ✓
\quadd by yark ✓
grammar and linking by Wkbj79 ✓