nyuuzyou/wikis huggingface dataset:
Divisor Count of 17,796,126,877,482,329,126,051

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 051} = 48$
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:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 051 = 7^5 \times 29 \times 351 \, 121 \times 103 \, 987 \, 345 \, 177$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,051
Divisor Count of 17,796,126,877,482,329,126,052

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 052} = 48$
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:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 052 = 2^2 \times 3 \times 149 \times 991 \, 723 \times 10 \, 036 \, 160 \, 394 \, 373$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,052
Divisor Count of 17,796,126,877,482,329,126,053

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 053} = 8$
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:
:$17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 053 = 449 \times 11 \, 618 \, 801 \times 3 \, 411 \, 283 \, 698 \, 997$
Thus:
Category:Examples of Divisor Count Function
Category:17,796,126,877,482,329,126,053
Divisor Count of 171

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {171} = 6$
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:
:$171 = 3^2 \times 19$
Thus:
The divisors of $171$ can be enumerated as:
:$1, 3, 9, 19, 57, 171$
Category:Examples of Divisor Count Function
Category:171
Divisor Count of 172

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {172} = 6$
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:
:$172 = 2^2 \times 43$
Thus:
The divisors of $172$ can be enumerated as:
:$1, 2, 4, 43, 86, 172$
Category:Examples of Divisor Count Function
Category:172
Divisor Count of 18

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {18} = 6$
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:
:$18 = 2 \times 3^2$
Thus:
The divisors of $18$ can be enumerated as:
:$1, 2, 3, 6, 9, 18$
Category:Examples of Divisor Count Function
Category:18
Divisor Count of 180

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {180} = 18$
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:
:$180 = 2^2 \times 3^2 \times 5$
Thus:
The divisors of $180$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180$
Category:Examples of Divisor Count Function
Category:180
Divisor Count of 190

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {190} = 8$
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:
:$190 = 2 \times 5 \times 19$
Thus:
The divisors of $190$ can be enumerated as:
:$1, 2, 5, 10, 19, 38, 95, 190$
Category:Examples of Divisor Count Function
Category:190
Divisor Count of 192

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {192} = 14$
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:
:$192 = 2^6 \times 3$
Thus:
The divisors of $192$ can be enumerated as:
:$1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192$
Category:Examples of Divisor Count Function
Category:192
Divisor Count of 2

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} 2 = 2$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
We have that $2$ is a prime number.
From Divisor Count Function of Prime Number:
:$\map {\sigma_0} 2 = 2$
Category:Examples of Divisor Count Function
Category:2
Divisor Count of 20

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {20} = 6$
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:
:$20 = 2^2 \times 5$
Thus:
The divisors of $20$ can be enumerated as:
:$1, 2, 4, 5, 10, 20$
Category:Examples of Divisor Count Function
Category:20
Divisor Count of 2025

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {2025} = 15$
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:
:$2025 = 3^4 \times 5^2$
Thus:
The divisors of $2025$ can be enumerated as:
:$1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025$
Category:Examples of Divisor Count Function
Category:2025
Divisor Count of 21,952

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {21 \, 952} = 28$
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:
:$21 \, 952 = 2^6 \times 7^3$
Thus:
The divisors of $21 \, 952$ can be enumerated as:
:$1, 2, 4, 7, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 196, 224, 343, 392,$
:$448, 686, 784, 1372, 1568, 2744, 3136, 5488, 10976, 21952$
Category:Examples of Divisor Count Function
Category:21,952
Divisor Count of 210

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {210} = 16$
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:
:$210 = 2 \times 3 \times 5 \times 7$
Thus:
The divisors of $210$ can be enumerated as:
:$1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210$
Category:Examples of Divisor Count Function
Category:210
Divisor Count of 2205

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {2205} = 18$
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:
:$2205 = 3^2 \times 5 \times 7^2$
Thus:
The divisors of $2205$ can be enumerated as:
:$1, 3, 5, 7, 9, 15, 21, 35, 45, 49, 63, 105, 147, 245, 315, 441, 735, 2205$
Category:Examples of Divisor Count Function
Category:2205
Divisor Count of 224

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {224} = 12$
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:
:$224 = 2^5 \times 7$
Thus:
The divisors of $224$ can be enumerated as:
:$1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 112, 224$
Category:Examples of Divisor Count Function
Category:224
Divisor Count of 24

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {24} = 8$
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:
:$24 = 2^3 \times 3$
Thus:
The divisors of $24$ can be enumerated as:
:$1, 2, 3, 4, 6, 8, 12, 24$
Category:Examples of Divisor Count Function
Category:24
Divisor Count of 240

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {240} = 20$
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:
:$240 = 2^4 \times 3 \times 5$
Thus:
The divisors of $240$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240$
Category:Examples of Divisor Count Function
Category:240
Divisor Count of 242

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {242} = 6$
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:
:$242 = 2 \times 11^2$
Thus:
The divisors of $242$ can be enumerated as:
:$1, 2, 11, 22, 121, 242$
Category:Examples of Divisor Count Function
Category:242
Divisor Count of 243

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {243} = 6$
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:
:$243 = 3^5$
Thus:
The divisors of $243$ can be enumerated as:
:$1, 3, 9, 27, 81, 243$
Category:Examples of Divisor Count Function
Category:243
Divisor Count of 244

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {244} = 6$
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:
:$244 = 2^2 \times 61$
Thus:
The divisors of $244$ can be enumerated as:
:$1, 2, 4, 61, 122, 244$
Category:Examples of Divisor Count Function
Category:244
Divisor Count of 245

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {245} = 6$
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:
:$245 = 5 \times 7^2$
Thus:
The divisors of $245$ can be enumerated as:
:$1, 5, 7, 35, 49, 245$
Category:Examples of Divisor Count Function
Category:245
Divisor Count of 248

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {248} = 8$
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:
:$248 = 2^3 \times 31$
Thus:
The divisors of $248$ can be enumerated as:
:$1, 2, 4, 8, 31, 62, 124, 248$
Category:Examples of Divisor Count Function
Category:248
Divisor Count of 2520

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {2520} = 48$
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:
:$2520 = 2^3 \times 3^2 \times 5 \times 7$
Thus:
The divisors of $2520$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72,$
:$84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520$
Category:Examples of Divisor Count Function
Category:2520
Divisor Count of 264

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {264} = 16$
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:
:$264 = 2^3 \times 3 \times 11$
Thus:
The divisors of $264$ can be enumerated as:
:$1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264$
Category:Examples of Divisor Count Function
Category:264
Divisor Count of 270

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {270} = 16$
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:
:$270 = 2 \times 3^3 \times 5$
Thus:
The divisors of $270$ can be enumerated as:
:$1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270$
Category:Examples of Divisor Count Function
Category:270
Divisor Count of 272

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {272} = 10$
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:
:$272 = 2^4 \times 17$
Thus:
The divisors of $272$ can be enumerated as:
:$1, 2, 4, 8, 16, 17, 34, 68, 136, 272$
Category:Examples of Divisor Count Function
Category:272
Divisor Count of 28

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {28} = 6$
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:
:$28 = 2^2 \times 7$
Thus:
The divisors of $28$ can be enumerated as:
:$1, 2, 4, 7, 14, 28$
Category:Examples of Divisor Count Function
Category:28
Divisor Count of 30

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {30} = 8$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
From Divisor Count Function from Prime Decomposition:
:$\map {\sigma_0} n = \ds \prod_{j \mathop = 1}^r \paren {k_j + 1}$
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:
:$30 = 2 \times 3 \times 5$
Thus:
The divisors of $30$ can be enumerated as:
:$1, 2, 3, 5, 6, 10, 15, 30$
Category:Examples of Divisor Count Function
Category:30
Divisor Count of 304

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {304} = 10$
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:
:$304 = 2^4 \times 19$
Thus:
The divisors of $304$ can be enumerated as:
:$1, 2, 4, 8, 16, 19, 38, 76, 152, 304$
Category:Examples of Divisor Count Function
Category:304
Divisor Count of 32

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {32} = 6$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
The divisors of $32$ can be enumerated as:
:$1, 2, 4, 8, 16, 32$
Category:Examples of Divisor Count Function
Category:32
Divisor Count of 332

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {332} = 6$
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:
:$332 = 2^2 \times 83$
Thus:
The divisors of $332$ can be enumerated as:
:$1, 2, 4, 83, 166, 332$
Category:Examples of Divisor Count Function
Category:332
Divisor Count of 333

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {333} = 6$
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:
:$333 = 3^2 \times 37$
Thus:
The divisors of $333$ can be enumerated as:
:$1, 3, 9, 37, 111, 333$
Category:Examples of Divisor Count Function
Category:333
Divisor Count of 35

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {35} = 4$
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:
:$35 = 5 \times 7$
Thus:
The divisors of $35$ can be enumerated as:
:$1, 5, 7, 35$
Category:Examples of Divisor Count Function
Category:35
Divisor Count of 350

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {350} = 12$
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:
:$350 = 2 \times 5^2 \times 7$
Thus:
The divisors of $350$ can be enumerated as:
:$1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350$
Category:Examples of Divisor Count Function
Category:350
Divisor Count of 357

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {357} = 8$
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:
:$357 = 3 \times 7 \times 17$
Thus:
The divisors of $357$ can be enumerated as:
:$1, 3, 7, 17, 21, 51, 119, 357$
Category:Examples of Divisor Count Function
Category:357
Divisor Count of 360

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {360} = 24$
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:
:$360 = 2^3 \times 3^2 \times 5$
Thus:
The divisors of $360$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360$
Category:Examples of Divisor Count Function
Category:360