nyuuzyou/wikis huggingface dataset:
Particular Values of Secant Function

== Theorem ==
The following values of the secant function can be expressed as exact algebraic numbers.
This list is non-exhaustive.
 
=== Secant of Zero ===
=== Secant of $15 \degrees$ ===
=== Secant of $30 \degrees$ ===
=== Secant of $45 \degrees$ ===
=== Secant of $60 \degrees$ ===
=== Secant of $75 \degrees$ ===
=== Secant of Right Angle ===
=== Secant of $105 \degrees$ ===
=== Secant of $120 \degrees$ ===
=== Secant of $135 \degrees$ ===
=== Secant of $150 \degrees$ ===
=== Secant of $165 \degrees$ ===
=== Secant of Straight Angle ===
=== Secant of $195 \degrees$ ===
=== Secant of $210 \degrees$ ===
=== Secant of $225 \degrees$ ===
=== Secant of $240 \degrees$ ===
=== Secant of $255 \degrees$ ===
=== Secant of Three Right Angles ===
=== Secant of $285 \degrees$ ===
=== Secant of $300 \degrees$ ===
=== Secant of $315 \degrees$ ===
=== Secant of $330 \degrees$ ===
=== Secant of $345 \degrees$ ===
=== Secant of Full Angle ===
== Also see ==
* Particular Values of Sine Function
* Particular Values of Cosine Function
* Particular Values of Tangent Function
* Particular Values of Cotangent Function
* Particular Values of Cosecant Function
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Secant Function
Particular Values of Signed Stirling Numbers of the First Kind

== Theorem ==
This page gathers together some particular values of signed Stirling numbers of the first kind.
=== Signed Stirling Number of the First Kind: $\map s {0, n}$ ===
=== Signed Stirling Number of the First Kind: $\map s {1, n}$ ===
=== Signed Stirling Number of the First Kind: $\map s {n, n}$ ===
=== Signed Stirling Number of the First Kind: $\map s {n, n - 1}$ ===
=== Signed Stirling Number of the First Kind: $\map s {n + 1, 0}$ ===
=== Signed Stirling Number of the First Kind: $\map s {n + 1, 1}$ ===
== Also see ==
* Particular Values of Unsigned Stirling Numbers of the First Kind
* Particular Values of Stirling Numbers of the Second Kind
Category:Stirling Numbers
Category:Examples of Stirling Numbers of the First Kind
Particular Values of Sine Function

== Theorem ==
The following values of the sine function can be expressed as exact algebraic numbers.
This list is non-exhaustive.
 
=== Sine of Zero ===
=== Sine of 15 Degrees ===
=== Sine of 30 Degrees ===
=== Sine of 45 Degrees ===
=== Sine of 60 Degrees ===
=== Sine of 75 Degrees ===
=== Sine of Right Angle ===
=== Sine of 105 Degrees ===
=== Sine of 120 Degrees ===
=== Sine of 135 Degrees ===
=== Sine of 150 Degrees ===
=== Sine of 165 Degrees ===
=== Sine of Straight Angle ===
=== Sine of 195 Degrees ===
=== Sine of 210 Degrees ===
=== Sine of 225 Degrees ===
=== Sine of 240 Degrees ===
=== Sine of 255 Degrees ===
=== Sine of Three Right Angles ===
=== Sine of 285 Degrees ===
=== Sine of 300 Degrees ===
=== Sine of 315 Degrees ===
=== Sine of 330 Degrees ===
=== Sine of 345 Degrees ===
=== Sine of Full Angle ===
== Also see ==
* Particular Values of Cosine Function
* Particular Values of Tangent Function
* Particular Values of Cotangent Function
* Particular Values of Secant Function
* Particular Values of Cosecant Function
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Sine Function
Particular Values of Stirling Numbers of the Second Kind

== Theorem ==
This page gathers together some particular values of Stirling numbers of the second kind.
=== Stirling Number of the Second Kind: $\ds {0 \brace n}$ ===
=== Stirling Number of the Second Kind: $\ds {1 \brace n}$ ===
=== Stirling Number of the Second Kind: $\ds {n \brace n}$ ===
=== Stirling Number of the Second Kind: $\ds {n \brace n - 1}$ ===
=== Stirling Number of the Second Kind: $\ds {n \brace n - 2}$ ===
=== Stirling Number of the Second Kind: $\ds {n \brace n - 3}$ ===
=== Stirling Number of the Second Kind: $\ds {n + 1 \brace 0}$ ===
=== Stirling Number of the Second Kind: $\ds {n + 1 \brace 1}$ ===
=== Stirling Number of the Second Kind: $\ds {n + 1 \brace 2}$ ===
== Also see ==
* Particular Values of Unsigned Stirling Numbers of the First Kind
* Particular Values of Signed Stirling Numbers of the First Kind
Category:Stirling Numbers
Category:Examples of Stirling Numbers of the Second Kind
Particular Values of Tangent Function

== Theorem ==
The following values of the tangent function can be expressed as exact algebraic numbers.
This list is non-exhaustive.
 
=== Tangent of Zero ===
=== Tangent of 15 Degrees ===
=== Tangent of 22.5 Degrees ===
=== Tangent of 30 Degrees ===
=== Tangent of 45 Degrees ===
=== Tangent of 60 Degrees ===
=== Tangent of 67.5 Degrees ===
=== Tangent of 75 Degrees ===
=== Tangent of Right Angle ===
=== Tangent of 105 Degrees ===
=== Tangent of 120 Degrees ===
=== Tangent of 135 Degrees ===
=== Tangent of 150 Degrees ===
=== Tangent of 165 Degrees ===
=== Tangent of Straight Angle ===
=== Tangent of 195 Degrees ===
=== Tangent of 210 Degrees ===
=== Tangent of 225 Degrees ===
=== Tangent of 240 Degrees ===
=== Tangent of 255 Degrees ===
=== Tangent of Three Right Angles ===
=== Tangent of 285 Degrees ===
=== Tangent of 300 Degrees ===
=== Tangent of 315 Degrees ===
=== Tangent of 330 Degrees ===
=== Tangent of 345 Degrees ===
=== Tangent of Full Angle ===
== Also see ==
* Particular Values of Sine Function
* Particular Values of Cosine Function
* Particular Values of Cotangent Function
* Particular Values of Secant Function
* Particular Values of Cosecant Function
== Sources ==
* : $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles
Category:Tangent Function
Particular Values of Unsigned Stirling Numbers of the First Kind

== Theorem ==
This page gathers together some particular values of unsigned Stirling numbers of the first kind.
=== Unsigned Stirling Number of the First Kind: $\ds {0 \brack n}$ ===
=== Unsigned Stirling Number of the First Kind: $\ds {1 \brack n}$ ===
=== Unsigned Stirling Number of the First Kind: $\ds {n \brack n}$ ===
=== Unsigned Stirling Number of the First Kind: $\ds {n \brack n - 1}$ ===
=== Unsigned Stirling Number of the First Kind: $\ds {n \brack n - 2}$ ===
=== Unsigned Stirling Number of the First Kind: $\ds {n \brack n - 3}$ ===
=== Unsigned Stirling Number of the First Kind: $\ds {n + 1 \brack 0}$ ===
=== Unsigned Stirling Number of the First Kind: $\ds {n + 1 \brack 1}$ ===
== Also see ==
* Particular Values of Signed Stirling Numbers of the First Kind
* Particular Values of Stirling Numbers of the Second Kind
Category:Stirling Numbers
Category:Examples of Stirling Numbers of the First Kind
Partition Equation

== Theorem ==
Let group $G$ act on a finite set $X$.
Let the distinct orbits of $X$ under the action of $G$ be:
:$\Orb {x_1}, \Orb {x_2}, \ldots, \Orb {x_s}$
Then:
:$\card X = \card {\Orb {x_1} } + \card {\Orb {x_2} } + \cdots + \card {\Orb {x_s} }$
== Proof ==
Follows trivially from the fact that the Group Action Induces Equivalence Relation.
Category:Group Actions
Category:Named Theorems
Partition Space is Pseudometrizable

== Theorem ==
Let $T = \struct {S, \tau}$ be a partition space.
Then $T$ is pseudometrizable.
== Proof ==
Let $\PP$ be the partition which is the basis for $T$.
Let us define $d: S^2 \to \R$ by:
:$\forall x, y \in S: \map d {x, y} = \begin{cases}
0 & : \exists U \in \PP: x, y \in U \\
1 & : \text{otherwise} \end{cases}$
That is, $\map d {x, y} = 0$ when $x$ and $y$ are in the same set in the partition $\PP$, and $1$ otherwise.
This is easily established as being a pseudometric on $S$.
Hence the result.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology: $6$
Category:Pseudometrizable Topologies
Category:Partition Topology
Partition Topology is T3

== Corollary to Partition Topology is T3 1/2 ==
Let $S$ be a set and let $\PP$ be a partition on $S$.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then:
:$T$ is a $T_3$ space.
== Proof ==
We have that the Partition Topology is $T_{3 \frac 1 2}$.
We also have that a $T_{3 \frac 1 2}$ Space is $T_3$ Space.
The result follows.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$
Category:T3 Spaces
Category:Partition Topology
Partition Topology is T3 1/2

== Theorem ==
Let $S$ be a set and let $\PP$ be a partition on $S$.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is a $T_{3 \frac 1 2}$ space.
=== Corollary ===
== Proof ==
Let $F \subseteq S$ be closed.
Denote by $S \setminus F$ the relative complement of $F$ in $S$.
Let $x \in S \setminus F$.
Define a mapping $f: S \to \closedint 0 1$ as:
:$\map f s := \begin{cases}
 1 & : \text { if } s \in F \\
 0 & : \text { if } s \in S \setminus F
\end{cases}$
Then $f$ is identically $1$ on $F$, and identically $0$ on $\set x$.
Now if $f$ is continuous, it will be a Urysohn function for $F$ and $\set y$, and $T$ will be a $T_{3 \frac 1 2}$ space.
Now for any $V \subseteq \closedint 0 1$, we have:
:$f^{-1} \sqbrk V = \begin{cases}
 \O & : \text{ if } 0, 1 \notin V \\
 F & : \text{ if } 0 \notin V \text { and } 1 \in V \\
 S \setminus F & : \text{ if } 0 \in V \text { and } 1 \notin V \\
 S & : \text{ if } 0, 1 \in V
\end{cases}$
By definition of $\tau$, $F$ is open in $T$.
By Open Set in Partition Topology is also Closed, $F$ is also closed, and so $S \setminus F$ is open in $T$.
Thus, the preimage of any subset $V$ of $\closedint 0 1$ is open in $T$.
In particular, this holds for the open sets of $\closedint 0 1$.
It follows that $f$ is a continuous mapping, and so a Urysohn function.
Hence $T$ is $T_{3 \frac 1 2}$ space.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$
Category:T3 1/2 Spaces
Category:Partition Topology
Partition Topology is T4

== Corollary to Partition Topology is T5 ==
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then:
:$T$ is a $T_4$ space.
== Proof ==
We have that the Partition Topology is $T_5$.
We also have that a $T_5$ Space is $T_4$ Space.
The result follows.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$
Category:T4 Spaces
Category:Partition Topology
Partition Topology is T5

== Theorem ==
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is a $T_5$ space.
=== Corollary ===
== Proof ==
Let $A$ and $B$ be subsets of $S$ such that $A^- \cap B = A \cap B^- = \O$.
From Open Set in Partition Topology is also Closed, we get that $A^-$ and $B^-$ are both clopen. 
From Set is Subset of its Topological Closure:
:$A \subseteq A^-$
From $A^- \cap B = \O$ it follows from Intersection with Complement is Empty iff Subset that $B \subseteq S \setminus \paren {A^-}$.
Since $A^-$ is clopen, it follows that $A^-$ and $S \setminus \paren {A^-}$ are both open.
Also, they are disjoint by definition.
Thus we have $A^-$, an open set containing $A$, and $S \setminus \paren {A^-}$, an open set containing $B$.
Thus, by definition, $T$ is a $T_5$ space.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$
Category:T5 Spaces
Category:Partition Topology
Partition Topology is Topology

== Theorem ==
Let $S$ be a set.
Let $\PP$ be a partition of $S$.
Let $\tau$ be the set of subsets of $S$ defined as:
:$a \in \tau \iff a$ is the union of sets of $\PP$
Then $\tau$ is a topology on $S$.
== Proof ==
From Basis for Partition Topology, we have that $\PP$ is a basis for the partition topology.
The result follows.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology
Category:Partition Topology
Partition Topology is Zero Dimensional

== Theorem ==
Let $T = \struct {S, \tau}$ be a partition space.
Then $T$ is zero dimensional.
== Proof ==
Let $\PP$ be the partition which is the basis for $T$.
From Open Set in Partition Topology is also Closed, all the elements of $\PP$ are both closed and open.
Hence the result, by definition of zero dimensional space
Category:Zero Dimensional Spaces
Category:Partition Topology
Partition Topology is not Completely Hausdorff

== Theorem ==
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is not a $T_{2 \frac 1 2}$ (completely Hausdorff) space.
== Proof ==
 $T$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space.
Then from Completely Hausdorff Space is Hausdorff Space, $T$ is a $T_2$ (Hausdorff) space.
This contradicts the result Partition Topology is not Hausdorff.
Hence the result, by Proof by Contradiction.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$
Category:Completely Hausdorff Spaces
Category:Partition Topology
Partition Topology is not Hausdorff

== Theorem ==
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is not a $T_2$ (Hausdorff) space.
== Proof ==
 $T$ is a $T_2$ (Hausdorff) space.
Then from $T_2$ Space is $T_1$ Space, $T$ is a $T_1$ (Fréchet) space.
This contradicts the result Partition Topology is not $T_1$.
Hence the result, by Proof by Contradiction.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$
Category:Hausdorff Spaces
Category:Partition Topology
Partition Topology is not T0

== Theorem ==
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is not a $T_0$ (Kolmogorov) space.
== Proof ==
As $\PP$ is not the partition of singletons, there exists some $H \in \PP$ such that $a, b \in H: a \ne b$.
Any union of sets from $\PP$ which includes $H$ will therefore contain both $a$ and $b$.
Therefore, any element of $\tau$ containing $a$ will also contain $b$, and similarly, any element of $\tau$ containing $b$ will also contain $a$.
So there is no open set in $T$ containing $a$ and not $b$, or $b$ and not $a$.
Hence the result, by definition of $T_0$ (Kolmogorov) space.
Note the reason for the exception in the statement as given.
If $\PP$ is the partition of singletons:
:$\PP = \set {\set x: x \in S}$
then $T$ as constructed is the discrete topology on $S$.
From Discrete Space satisfies all Separation Properties, in that case $T$ does satisfy the $T_0$ property.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$
Category:T0 Spaces
Category:Partition Topology
Partition Topology is not T1

== Theorem ==
Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is not a $T_1$ (Fréchet) space.
== Proof ==
 $T$ is a $T_1$ (Fréchet) space.
Then from $T_1$ Space is $T_0$ Space, $T$ is a $T_0$ (Kolmogorov) space.
This contradicts the result Partition Topology is not $T_0$.
Hence the result, by Proof by Contradiction.
== Sources ==
* : Part $\text {II}$: Counterexamples: $5$. Partition Topology: $2$
Category:T1 Spaces
Category:Partition Topology
Partition of Facets of Rubik's Cube

== Theorem ==
Let $S$ denote the set of the facets of Rukik's cube.
Then $S$ can be partitioned as follows:
:$S = \set {S_C \mid S_E \mid S_Z}$
where:
:$S_C$ denotes the set of corner facets
:$S_E$ denotes the set of edge facets
:$S_Z$ denotes the set of center facets.
== Proof ==
From the definition of the facets, each face is divided into $9$ facets.
:
A facet is either:
:on the corner of a face, for example $flu$, $fru$
:on the edge of a face, for example $fu$, $fr$
:in the center of a face, for example $F$.
:$(1):\quad$ Each facet can be either in $S_C$ or $S_E$ or $S_Z$ and can not be in more than one.
:$(2):\quad$ Each facet can be either in $S_C$ or $S_E$ or $S_Z$ and there are no other possibilities.
:$(3):\quad$ None of $S_C$, $S_E$ and $S_Z$ is empty.
Thus the criteria for $S = \set {S_C \mid S_E \mid S_Z}$ to be a partition are fulfilled.
== Sources ==
* : Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson: Example $1.2.3$
Category:Rubik's Cube
Partition of Indexing Set induces Bijection on Family of Sets

== Theorem ==
Let $I$ be an indexing set.
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.
Let $\family {I_\gamma}_{\gamma \mathop \in J}$ be a partitioning of $I$.
Then there exists a bijection:
:$\ds \phi: \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} S_\alpha} \to \prod_{\alpha \mathop \in I} S_\alpha$
== Proof ==
First a lemma:
We can define a projection $\pr_\gamma$:
:$\ds \pr_\gamma: \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} S_\alpha} \to \prod_{\alpha \mathop \in I_\gamma} S_\alpha$
so that for $\ds X \in \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} S_\alpha}, X = \family {X_\gamma}_{\gamma \mathop \in J}$:
:$\map {\pr_\gamma} X = X_\gamma$
From the lemma we can build $\phi$ so that for $I_\gamma = \set {\alpha: \alpha \in I_\gamma}$:
:$\ds \map \phi {\mathbf x} = \prod_{\gamma \mathop \in J} \paren {\prod_{\alpha \mathop \in I_\gamma} \map {\psi_\alpha} {x_\alpha} }$
as $\phi$ uniquely sets all components of $\map {\pr_\gamma} X$ for all $\gamma$.
The result follows.
== Sources ==
* : Chapter $1$: Theory of Sets: $\S 10$: Arbitrary Products: Exercise $3$
Category:Indexed Families
Category:Bijections
Category:Set Partitions
Partition of Indexing Set induces Bijection on Family of Sets/Lemma

== Theorem ==
Let $I$ be an indexing set.
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.
Let $I = I_1 \cup I_2$ such that $I_1 \cap I_2 = \O$.
Then there exists a bijection:
:$\ds \psi: \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha} \to \prod_{\alpha \mathop \in I} S_\alpha$
== Proof ==
Let us define the mapping:
:$\ds \psi: \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha} \to \prod_{\alpha \mathop \in I} S_\alpha$
$\psi$ can be injective :
:$\map \psi a = \map \psi {a'} \implies a = a'$
where $\ds a, a' \in \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha}$
We can define a projection $\pr_1$:
:$\ds \pr_1: \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha} \to \prod_{\alpha \mathop \in I_1} S_\alpha$
so that for $\ds X \in \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha}, X = \tuple {X_1, X_2}$:
:$\map {\pr_1} X = X_1$
and similarly a projection $\pr_2$:
:$\ds \pr_2: \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha} \to \prod_{\alpha \mathop \in I_2} S_\alpha$
so that for $\ds X \in \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha}, X = \tuple {X_1, X_2}$:
:$\map {\pr_2} X = X_2$
From these we can build $\psi$ so that for $I_1 = \set {\alpha: \alpha \in I_1}$ and $I_2 = \set {\alpha: \alpha \in I_2}$:
:$\ds \map \psi {\mathbf x} = \paren {\prod_{\alpha \mathop \in I_1} \map {\psi_\alpha} {x_\alpha} } \times \paren {\prod_{\alpha \mathop \in I_2} \map {\psi_\alpha} {x_\alpha} }$
as $\psi$ uniquely sets all components of $\map {\pr_1} X$ and $\map {\pr_2} X$.
These in turn uniquely set the $2$ components of $X$.
That is:
:$\map f X = \map f {X'} \implies X = X'$
Surjection follows trivially.
== Sources ==
* : Chapter $1$: Theory of Sets: $\S 10$: Arbitrary Products: Exercise $3$
Category:Indexed Families
Category:Bijections
Category:Set Partitions
Partition of Integer into Odd Parts/Examples/5

== Examples of Partition of Integer into Odd Parts ==
The integer $5$ can be partitioned into odd parts as follows:
:$5$
:$3 + 1 + 1$
:$1 + 1 + 1 + 1 + 1$
== Sources ==
* : $\text {3-5}$ The Use of Computers in Number Theory: Exercise $9$
Category:Examples of Integer Partitions
Partition of Integer into Odd Parts/Table

== Table of Integer Partitions ==
The following table presents a list of the number of ways a positive integer can be partitioned into odd parts for all $n$ from $1$ to $30$.
In the following, $\map p n$ denotes the number of integer partitions into odd parts for $n$.
:$\begin{array} {|r|r|}
\hline
n & \map p n \\
\hline
1 & 1 \\
2 & 1 \\
3 & 2 \\
4 & 2 \\
5 & 3 \\
6 & 4 \\
7 & 5 \\
8 & 6 \\
9 & 8 \\
10 & 10 \\
11 & 12 \\
12 & 15 \\
13 & 18 \\
14 & 22 \\
15 & 27 \\
16 & 32 \\
17 & 38 \\
18 & 46 \\
19 & 54 \\
20 & 64 \\
21 & 76 \\
22 & 89 \\
23 & 104 \\
24 & 122 \\
25 & 142 \\
26 & 165 \\
27 & 192 \\
28 & 222 \\
29 & 256 \\
30 & 296 \\
\hline
\end{array}$,
== Sources ==
* : $\text {3-5}$ The Use of Computers in Number Theory: Exercise $9$
Category:Examples of Integer Partitions
Partition of Integer into Parts not Multiple of 3/Examples/4

== Examples of Partition of Integer into Parts not Multiple of 3 ==
The integer $4$ can be partitioned into parts which are specifically not multiples of $3$ as follows:
:$4$
:$2 + 2$
:$2 + 1 + 1$
:$1 + 1 + 1 + 1$
== Sources ==
* : $\text {3-5}$ The Use of Computers in Number Theory: Exercise $13$
Category:Examples of Integer Partitions
Partition of Integer into Parts not Multiple of 3/Table

== Table of Partition of Integer into Parts not Multiple of 3 ==
The following table presents a list of the number of ways a positive integer $n$ can be partitioned into parts which are specifically not multiples of $3$ for all $n$ from $1$ to $15$.
In the following, $\map t n$ denotes the number of such partitions for $n$.
:$\begin{array} {|r|r|}
\hline
n & \map t n \\
\hline
1 & 1 \\
2 & 2 \\
3 & 2 \\
4 & 4 \\
5 & 5 \\
6 & 7 \\
7 & 9 \\
8 & 13 \\
9 & 16 \\
10 & 22 \\
11 & 27 \\
12 & 36 \\
13 & 44 \\
14 & 57 \\
15 & 70 \\
\hline
\end{array}$
== Sources ==
* : $\text {3-5}$ The Use of Computers in Number Theory: Exercise $13$
Category:Examples of Integer Partitions
Partition of Integer into Powers of 2/Examples/4

== Example of Partition of Integer into Powers of 2 ==
The integer $4$ can be partitioned into (integer) powers of $2$ as follows:
:$4$
:$2 + 2$
:$2 + 1 + 1$
:$1 + 1 + 1 + 1$
== Sources ==
* : $\text {3-5}$ The Use of Computers in Number Theory: Exercise $12$
Category:Examples of Integer Partitions
Partition of Integer into Powers of 2/Table

== Table of Partition of Integer into Powers of 2 ==
The following table presents a list of the number of ways a positive integer $n$ can be partitioned into (integer) powers of $2$ for all $n$ from $1$ to $40$.
In the following, $\map b n$ denotes the number of integer partitions into (integer) powers of $2$ for $n$.
:$\begin{array} {|r|r|}
\hline
n & \map b n \\
\hline
1 & 1 \\
2 & 2 \\
3 & 2 \\
4 & 4 \\
5 & 4 \\
6 & 6 \\
7 & 6 \\
8 & 10 \\
9 & 10 \\
10 & 14 \\
11 & 14 \\
12 & 20 \\
13 & 20 \\
14 & 26 \\
15 & 26 \\
16 & 36 \\
17 & 36 \\
18 & 46 \\
19 & 46 \\
20 & 60 \\
\hline
\end{array} \qquad
\begin{array} {|r|r|}
\hline
n & \map b n \\
\hline
21 & 60 \\
22 & 74 \\
23 & 74 \\
24 & 94 \\
25 & 94 \\
26 & 114 \\
27 & 114 \\
28 & 140 \\
29 & 140 \\
30 & 166 \\
31 & 166 \\
32 & 202 \\
33 & 202 \\
34 & 238 \\
35 & 238 \\
36 & 284 \\
37 & 284 \\
38 & 330 \\
39 & 330 \\
40 & 390 \\
\hline
\end{array}$ 
== Sources ==
* : $\text {3-5}$ The Use of Computers in Number Theory: Exercise $12$
Category:Examples of Integer Partitions