nyuuzyou/wikis huggingface dataset:
Divisor Count Function is Primitive Recursive

== Theorem ==
The divisor count function is primitive recursive.
== Proof ==
The divisor count function $\sigma_0: \N \to \N$ is defined as:
:$\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$
where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.
Thus we can define $\map {\sigma_0} n$ as:
:$\ds \map {\sigma_0} n = \sum_{y \mathop = 1}^n \map {\operatorname {div} } {n, y}$
where
:$\map {\operatorname {div} } {n, y} = \begin{cases}
1 & : y \divides n \\
0 & : y \nmid n
\end{cases}$
Hence $\sigma_0$ is defined by substitution from:
:the primitive recursive function $\operatorname {div}$
:the primitive recursive bounded summation $\ds \sum_{y \mathop = 1}^n$.
Hence the result.
Category:Primitive Recursive Functions
Category:Divisor Count Function
Divisor Count Function of Power of Prime

== Theorem ==
Let $n = p^k$ be the power of a prime number $p$.
Let $\map {\sigma_0} n$ be the divisor count function of $n$.
That is, let $\map {\sigma_0} n$ be the number of positive divisors of $n$.
Then:
:$\map {\sigma_0} n = k + 1$
== Proof ==
From Divisors of Power of Prime, the divisors of $n = p^k$ are:
:$1, p, p^2, \ldots, p^{k - 1}, p^k$
There are $k + 1$ of them.
Hence the result.
== Sources ==
* 
Category:Divisor Count Function
Category:Prime Numbers
Divisor Count Function of Prime Number

== Theorem ==
Let $p \in \Z_{> 0}$.
Then $p$ is a prime number :
:$\map {\sigma_0} p = 2$
where $\map {\sigma_0} p$ denotes the divisor count function of $p$.
== Proof ==
=== Necessary Condition ===
Let $p$ be a prime number.
Then, by definition, the only positive divisors of $p$ are $1$ and $p$.
Hence by definition of the divisor count function:
:$\map {\sigma_0} p = 2$
=== Sufficient Condition ===
Suppose $\map {\sigma_0} p = 2$.
Then by One Divides all Integers we have:
:$1 \divides p$
Also, by Integer Divides Itself we have:
:$p \divides p$
So if $p > 1$ it follows that $\map {\sigma_0} p \ge 2$.
Now for $\map {\sigma_0} p = 2$ it must follow that the only divisors of $p$ are $1$ and $p$.
That is, that $p$ is a prime number.
Category:Prime Numbers
Category:Divisor Count Function
Divisor Count of 102

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {102} = 8$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$102 = 2 \times 3 \times 17$
Thus:
The divisors of $102$ can be enumerated as:
:$1, 2, 3, 6, 17, 34, 51, 102$
Category:Examples of Divisor Count Function
Category:102
Divisor Count of 104

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {104} = 8$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$104 = 2^3 \times 13$
Thus:
The divisors of $104$ can be enumerated as:
:$1, 2, 4, 8, 13, 26, 52, 104$
Category:Examples of Divisor Count Function
Category:104
Divisor Count of 108/Mistake

== Source Work ==
:
:The Dictionary
::$108$
== Mistake ==
:The number of its factors is also a cube: $8$.
Notwithstanding the fact that the also is redundant, as $108$ is not actually a cube itself for $8$ to be also to, the count of the factors of $108$ is in fact $12$, not $8$.
This is shown at .
It is probable that this is a typesetting error, as in the following paragraph it is pointed out that $\map {\sigma_1} {110}$ is a cube.
We also have from that $\map {\sigma_0} {110}$ is indeed equal to $8$.
== Sources ==
* : $108$
Category:Mistakes/Curious and Interesting Numbers
Divisor Count of 111,111

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {111 \, 111} = 32$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$111 \, 111 = 3 \times 7 \times 11 \times 13 \times 37$
Thus:
The divisors of $111 \, 111$ can be enumerated as:
:$1, 3, 7, 11, 13, 21, 33, 37, 39, 77, 91, 111, 143, 231,$
:$259, 273, 407, 429, 481, 777, 1001, 1221, 1443, 2849,$
:$3003, 3367, 5291, 8547, 10101, 15873, 37037, 111111$
Category:Examples of Divisor Count Function
Category:111,111
Divisor Count of 116

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {116} = 6$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$116 = 2^2 \times 29$
Thus:
The divisors of $116$ can be enumerated as:
:$1, 2, 4, 29, 58, 116$
Category:Examples of Divisor Count Function
Category:116
Divisor Count of 117

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {117} = 6$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$117 = 3^2 \times 13$
Thus:
The divisors of $117$ can be enumerated as:
:$1, 3, 9, 13, 39, 117$
Category:Examples of Divisor Count Function
Category:117
Divisor Count of 1184

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1184} = 12$
where ${\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1184 = 2^5 \times 37$
Thus:
The divisors of $1184$ can be enumerated as:
:$1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184$
Category:Examples of Divisor Count Function
Category:1184
Divisor Count of 1190

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1190} = 16$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1190 = 2 \times 5 \times 7 \times 17$
Thus:
The divisors of $1190$ can be enumerated as:
:$1, 2, 5, 7, 10, 14, 17, 34, 35, 70, 85, 119, 170, 238, 595, 1190$
Category:Examples of Divisor Count Function
Category:1190
Divisor Count of 1260

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1260} = 36$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1260 = 2^2 \times 3^2 \times 5 \times 7$
Thus:
The divisors of $1260$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260$
Category:Examples of Divisor Count Function
Category:1260
Divisor Count of 1312

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1312} = 12$
where ${\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1312 = 2^5 \times 41$
Thus:
The divisors of $1312$ can be enumerated as:
:$1, 2, 4, 8, 16, 32, 41, 82, 164, 328, 656, 1312$
Category:Examples of Divisor Count Function
Category:1312
Divisor Count of 1330

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1330} = 16$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1330 = 2 \times 5 \times 7 \times 19$
Thus:
The divisors of $1330$ can be enumerated as:
:$1, 2, 5, 7, 10, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330$
Category:Examples of Divisor Count Function
Category:1330
Divisor Count of 1376

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1376} = 12$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1376 = 2^5 \times 43$
Thus:
The divisors of $1376$ can be enumerated as:
:$1, 2, 4, 8, 16, 32, 43, 86, 172, 344, 688, 1376$
Category:Examples of Divisor Count Function
Category:1376
Divisor Count of 14

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {14} = 4$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$14 = 2 \times 7$
Thus:
The divisors of $14$ can be enumerated as:
:$1, 2, 7, 14$
Category:Examples of Divisor Count Function
Category:14
Divisor Count of 140

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {140} = 12$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$140 = 2^2 \times 5 \times 7$
Thus:
The divisors of $140$ can be enumerated as:
:$1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140$
Category:Examples of Divisor Count Function
Category:140
Divisor Count of 1430

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1430} = 16$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1430 = 2 \times 5 \times 11 \times 13$
Thus:
The divisors of $1430$ can be enumerated as:
:$1, 2, 5, 10, 11, 13, 22, 26, 55, 65, 110, 130, 143, 286, 715, 1430$
Category:Examples of Divisor Count Function
Category:1430
Divisor Count of 147

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {147} = 6$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$147 = 3 \times 7^2$
Thus:
The divisors of $147$ can be enumerated as:
:$1, 3, 7, 21, 49, 147$
Category:Examples of Divisor Count Function
Category:147
Divisor Count of 148

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {148} = 6$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$148 = 2^2 \times 37$
Thus:
The divisors of $148$ can be enumerated as:
:$1, 2, 4, 37, 74, 148$
Category:Examples of Divisor Count Function
Category:148
Divisor Count of 1485

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1485} = 16$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1485 = 3^3 \times 5 \times 11$
Thus:
The divisors of $1485$ can be enumerated as:
:$1, 3, 5, 9, 11, 15, 27, 33, 45, 55, 99, 135, 165, 297, 495, 1485$
Category:Examples of Divisor Count Function
Category:1485
Divisor Count of 15

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {15} = 4$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$15 = 3 \times 5$
Thus:
The divisors of $15$ can be enumerated as:
:$1, 3, 5, 15$
Category:Examples of Divisor Count Function
Category:15
Divisor Count of 1575

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1575} = 18$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1575 = 3^2 \times 5^2 \times 7$
Thus:
The divisors of $1575$ can be enumerated as:
:$1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 63, 75, 105, 175, 225, 315, 525, 1575$
Category:Examples of Divisor Count Function
Category:1575
Divisor Count of 1638

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1638} = 24$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1638 = 2 \times 3^2 \times 7 \times 13$
Thus:
The divisors of $1638$ can be enumerated as:
:$1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638$
Category:Examples of Divisor Count Function
Category:1638
Divisor Count of 168

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {168} = 16$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$168 = 2^3 \times 3 \times 7$
Thus:
The divisors of $168$ can be enumerated as:
:$1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168$
Category:Examples of Divisor Count Function
Category:168
Divisor Count of 1680

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {1680} = 40$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$1680 = 2^4 \times 3 \times 5 \times 7$
Thus:
The divisors of $1680$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60,$
:$70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680$
Category:Examples of Divisor Count Function
Category:1680
Divisor Count of 17,796,126,877,482,329,126,043

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 043} = 32$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 043 = 3 \times 73 \times 2381 \times 63 \, 678 \, 479 \times 535 \, 956 \, 203$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,043
Divisor Count of 17,796,126,877,482,329,126,044

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044} = 48$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044 = 2^2 \times 7 \times 4327 \times 456 \, 293 \times 321 \, 911 \, 699 \, 243$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,044
Divisor Count of 17,796,126,877,482,329,126,045

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 045} = 48$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 045 = 5 \times 17^2 \times 47 \times 53 \times 4 \, 944 \, 062 \, 119 \, 125 \, 691$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,045
Divisor Count of 17,796,126,877,482,329,126,046

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 046} = 48$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 046 = 2 \times 3^2 \times 179 \times 5171 \times 1 \, 068 \, 133 \, 213 \, 285 \, 183$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,046
Divisor Count of 17,796,126,877,482,329,126,047

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 047} = 48$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 047 = 11^5 \times 23 \times 107 \times 44 \, 900 \, 425 \, 217 \, 777$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,047
Divisor Count of 17,796,126,877,482,329,126,048

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 048} = 48$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 048 = 2^5 \times 19 \times 4 \, 590 \, 338 \, 339 \times 6 \, 376 \, 424 \, 429$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,048
Divisor Count of 17,796,126,877,482,329,126,049

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 049} = 48$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 049 = 3 \times 13^2 \times 241 \times 557 \times 261 \, 484 \, 106 \, 225 \, 711$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,049
Divisor Count of 17,796,126,877,482,329,126,050

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 050} = 48$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
:$r$ denotes the number of distinct prime factors in the prime decomposition of $n$
:$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
We have that:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 050 = 2 \times 3^2 \times 179 \times 5171 \times 1 \, 068 \, 133 \, 213 \, 285 \, 183$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,050