Euler's cototient function

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The number of cototatives of $\scriptstyle n$, called Euler's cototient function $\scriptstyle {\overline {\varphi }}(n)$, is the number of positive integers up to $\scriptstyle n$ which are noncoprime (not coprime) to $\scriptstyle n$.

${\rm {Cototient}}(n):=n-{\rm {Totient}}(n)$

${\overline {\varphi }}(n):=n-\varphi (n)$

${\overline {\varphi }}(n):=\sum _{{i=1} \atop {(n,i)>1}}^{n}i=\sum _{i=1}^{n}(1-\delta _{1}^{(n,i)}),$

where $\scriptstyle (n,i)$ is the greatest common divisor (GCD) of $\scriptstyle n$ and $\scriptstyle i$, and $\scriptstyle \delta _{i}^{j}$ is the Kronecker delta of $\scriptstyle i$ and $\scriptstyle j$.

$\scriptstyle {\overline {\varphi }}(n),\,1\,\leq \,n\,\leq \,257,$ where $\scriptstyle {\overline {\varphi }}(n)$ is Euler's cototient function (A051953)

180     ${\overline {\varphi }}(n)$
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Totatives and cototatives

The integers from 1 to $\scriptstyle n$ that are coprime to $\scriptstyle n$ are called the totatives of $\scriptstyle n$, while the integers from 1 to $\scriptstyle n$ that are noncoprime to $\scriptstyle n$ are called the cototatives of $\scriptstyle n$.

Thus, the set of totatives of $\scriptstyle n$ is (Cf. coprimality triangle)

$\{1\leq i\leq n~|~(n,i)=1\},$

while the set of cototatives of $\scriptstyle n$ is

$\{1\leq i\leq n~|~(n,i)>1\},$

where $\scriptstyle (n,i)$ is the greatest common divisor (GCD) of $\scriptstyle n$ and $\scriptstyle i.$

Euler's totient function is then the number of totatives of $\scriptstyle n$, while Euler's cototient function is the number of cototatives of $\scriptstyle n$.

Thus, Euler's totient function is the cardinality of the set of totatives of $\scriptstyle n$:

$\varphi (n):=|\{1\leq i\leq n~|~(n,i)=1\}|,$

while Euler's cototient function is the cardinality of the set of cototatives of $\scriptstyle n$:

${\overline {\varphi }}(n):=|\{1\leq i\leq n~|~(n,i)>1\}|.$

Generating function

Dirichlet generating function

The Dirichlet generating function of $\scriptstyle {\overline {\varphi }}(n)$ is

$D_{\{{\overline {\varphi }}(n)\}}(s)\equiv \sum _{n=1}^{\infty }{\frac {{\overline {\varphi }}(n)}{n^{s}}}=~?.$

Related functions

Iterated Euler cototient function

See Iterated Euler totient function.

Cototient summatory function

The cototient summatory function (partial sums of Euler's cototient function) is

${\overline {\Phi }}(n):=\sum _{k=1}^{n}{\overline {\varphi }}(k),\ n\geq 1,$

where $\scriptstyle {\overline {\varphi }}(n)$ is Euler's cototient function.

Sequences

A051953 Cototient(n) := n - phi(n) = n - A000010(n).

{0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, 1, 8, 7, 8, 1, 12, 1, 12, 9, 12, 1, 16, 5, 14, 9, 16, 1, 22, 1, 16, 13, 18, 11, 24, 1, 20, 15, 24, 1, 30, 1, 24, 21, 24, 1, 32, 7, 30, 19, 28, 1, 36, 15, 32, 21, 30, 1, 44, 1, 32, 27, ...}

See also

• Peter Luschny, Sequences related to Euler's totient function.

• Coprimality triangle

• Coprimality
• Totatives
• Euler's totient function (number of totatives of n)
• Totient summatory function
• Coprimorial or product of totatives of n
• Cototatives
• Noncoprimorial or product of cototatives of n

• Dedekind psi function
• Psi summatory function