nyuuzyou/wikis huggingface dataset:
Divisor Count of 3600

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {3600} = 45$
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:
:$3600 = 2^4 \times 3^2 \times 5^2$
Thus:
The divisors of $3600$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, 36, 40, 45, 48, 50, 60, 72, 75, 80, 90, 100,$
:$120, 144, 150, 180, 200, 225, 240, 300, 360, 400, 450, 600, 720, 900, 1200, 1800, 3600$
Category:Examples of Divisor Count Function
Category:3600
Divisor Count of 3655

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {3655} = 8$
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:
:$3655 = 5 \times 17 \times 43$
Thus:
The divisors of $3655$ can be enumerated as:
:$1, 5, 17, 44, 85, 215, 731, 3655$
Category:Examples of Divisor Count Function
Category:3655
Divisor Count of 3656

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {3656} = 8$
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:
:$3655 = 2^3 \times 457$
Thus:
The divisors of $3656$ can be enumerated as:
:$1, 2, 4, 8, 457, 914, 1828, 3656$
Category:Examples of Divisor Count Function
Category:3656
Divisor Count of 3657

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {3657} = 8$
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:
:$3657 = 3 \times 23 \times 53$
Thus:
The divisors of $3657$ can be enumerated as:
:$1, 3, 23, 53, 69, 159, 1219, 3657$
Category:Examples of Divisor Count Function
Category:3657
Divisor Count of 3658

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {3658} = 8$
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:
:$3658 = 2 \times 31 \times 59$
Thus:
The divisors of $3658$ can be enumerated as:
:$1, 2, 31, 59, 62, 118, 1829, 3658$
Category:Examples of Divisor Count Function
Category:3658
Divisor Count of 368

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {368} = 10$
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:
:$368 = 2^4 \times 23$
Thus:
The divisors of $368$ can be enumerated as:
:$1, 2, 4, 8, 16, 23, 46, 92, 184, 368$
Category:Examples of Divisor Count Function
Category:368
Divisor Count of 4,100,625

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {4 \, 100 \, 625} = 45$
where $\sigma_0$ denotes the divisor cunt 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:
:$4 \, 100 \, 625 = 5^4 \times 3^8$
Thus:
Category:Examples of Divisor Count Function
Category:4,100,625
Divisor Count of 40,310

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {40 \, 310} = 8$
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:
:$40 \, 310 = 2 \times 5 \times 29 \times 139$
Thus:
The divisors of $40 \, 310$ can be enumerated as:
:$1, 2, 5, 10, 29, 58, 139, 145, 278, 290, 695, 1390, 4031, 8062, 20 \, 155, 40 \, 310$
Category:Examples of Divisor Count Function
Category:40,310
Divisor Count of 40,311

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {40 \, 311} = 8$
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:
:$40 \, 311 = 3^3 \times 1493$
Thus:
The divisors of $40 \, 311$ can be enumerated as:
:$1, 3, 9, 27, 1493, 4479, 13 \, 437, 40 \, 311$
Category:Examples of Divisor Count Function
Category:40,311
Divisor Count of 40,312

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {40 \, 312} = 8$
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:
:$40 \, 312 = 2^3 \times 5039$
Thus:
The divisors of $40 \, 312$ can be enumerated as:
:$1, 2, 4, 8, 5039, 10 \, 078, 20 \, 156, 40 \, 312$
Category:Examples of Divisor Count Function
Category:40,312
Divisor Count of 40,313

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {40 \, 313} = 8$
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:
:$40 \, 313 = 7 \times 13 \times 443$
Thus:
The divisors of $40 \, 313$ can be enumerated as:
:$1, 7, 13, 91, 443, 3101, 5759, 40 \, 313$
Category:Examples of Divisor Count Function
Category:40,313
Divisor Count of 40,314

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {40 \, 314} = 8$
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:
:$40 \, 314 = 2 \times 3 \times 6719$
Thus:
The divisors of $40 \, 314$ can be enumerated as:
:$1, 2, 3, 6, 6719, 13 \, 438, 20 \, 157, 40 \, 314$
Category:Examples of Divisor Count Function
Category:40,314
Divisor Count of 40,315

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {40 \, 315} = 8$
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:
:$40 \, 315 = 5 \times 11 \times 733$
Thus:
The divisors of $40 \, 315$ can be enumerated as:
:$1, 5, 11, 55, 733, 3665, 8063, 40 \, 315$
Category:Examples of Divisor Count Function
Category:40,315
Divisor Count of 40,316

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {40 \, 316} = 6$
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:
:$40 \, 316 = 2^2 \times 10 \, 079$
Thus:
The divisors of $40 \, 316$ can be enumerated as:
:$1, 2, 4, 10 \, 079, 20 \, 158, 40 \, 316$
Category:Examples of Divisor Count Function
Category:40,316
Divisor Count of 4030

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {4030} = 16$
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:
:$4030 = 2 \times 5 \times 13 \times 31$
Thus:
The divisors of $4030$ can be enumerated as:
:$1, 2, 5, 10, 13, 26, 31, 62, 65, 130, 155, 310, 403, 806, 2015, 4030$
Category:Examples of Divisor Count Function
Category:4030
Divisor Count of 418

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {418} = 16$
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:
:$418 = 2 \times 11 \times 19$
Thus:
The divisors of $418$ can be enumerated as:
:$1, 2, 11, 19, 22, 38, 209, 418$
Category:Examples of Divisor Count Function
Category:418
Divisor Count of 42

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {42} = 8$
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:
:$42 = 2 \times 3 \times 7$
Thus:
The divisors of $42$ can be enumerated as:
:$1, 2, 3, 6, 7, 14, 21, 42$
Category:Examples of Divisor Count Function
Category:42
Divisor Count of 420

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {420} = 24$
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:
:$420 = 2^2 \times 3 \times 5 \times 7$
Thus:
The divisors of $420$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420$
Category:Examples of Divisor Count Function
Category:420
Divisor Count of 44

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {44} = 6$
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:
:$44 = 2^2 \times 11$
Thus:
The divisors of $44$ can be enumerated as:
:$1, 2, 4, 11, 22, 44$
Category:Examples of Divisor Count Function
Category:44
Divisor Count of 45

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {45} = 6$
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:
:$45 = 3^2 \times 5$
Thus:
The divisors of $45$ can be enumerated as:
:$1, 3, 5, 9, 15, 45$
Category:Examples of Divisor Count Function
Category:45
Divisor Count of 464

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {464} = 10$
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:
:$464 = 2^4 \times 29$
Thus:
The divisors of $464$ can be enumerated as:
:$1, 2, 4, 8, 16, 29, 58, 116, 232, 464$
Category:Examples of Divisor Count Function
Category:464
Divisor Count of 490

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {490} = 12$
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:
:$490 = 2 \times 5 \times 7^2$
Thus:
The divisors of $490$ can be enumerated as:
:$1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490$
Category:Examples of Divisor Count Function
Category:490
Divisor Count of 496

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {496} = 10$
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:
:$496 = 2^4 \times 31$
Thus:
The divisors of $496$ can be enumerated as:
:$1, 2, 4, 8, 16, 31, 62, 124, 248, 496$
Category:Examples of Divisor Count Function
Category:496
Divisor Count of 5040

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {5040} = 60$
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:
:$5040 = 2^4 \times 3^2 \times 5 \times 7$
Thus:
The divisors of $5040$ can be enumerated as:
:$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90,$
:$105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040$
Category:Examples of Divisor Count Function
Category:5040
Divisor Count of 52

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {52} = 4$
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:
:$52 = 2^2 \times 13$
Thus:
The divisors of $52$ can be enumerated as:
:$1, 2, 4, 13, 26, 52$
Category:Examples of Divisor Count Function
Category:52
Divisor Count of 550

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {550} = 12$
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:
:$550 = 2 \times 5^2 \times 11$
Thus:
The divisors of $550$ can be enumerated as:
:$1, 2, 5, 10, 11, 22, 25, 50, 55, 110, 275, 550$
Category:Examples of Divisor Count Function
Category:550
Divisor Count of 56

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {56} = 8$
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:
:$56 = 2^3 \times 7$
Thus:
The divisors of $56$ can be enumerated as:
:$1, 2, 4, 7, 8, 14, 28, 56$
Category:Examples of Divisor Count Function
Category:56
Divisor Count of 570

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {570} = 16$
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:
:$570 = 2 \times 3 \times 5 \times 19$
Thus:
The divisors of $570$ can be enumerated as:
:$1, 2, 3, 5, 6, 10, 15, 19, 30, 38, 57, 95, 114, 190, 285, 570$
Category:Examples of Divisor Count Function
Category:570
Divisor Count of 572

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {572} = 12$
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:
:$572 = 2^2 \times 11 \times 13$
Thus:
The divisors of $572$ can be enumerated as:
:$1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286, 572$
Category:Examples of Divisor Count Function
Category:572
Divisor Count of 6

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} 6 = 4$
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:
:$6 = 2 \times 3$
Thus:
The divisors of $6$ can be enumerated as:
:$1, 2, 3, 6$
== Sources ==
* 
Category:Examples of Divisor Count Function
Category:6
Divisor Count of 616

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {616} = 16$
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:
:$616 = 2^3 \times 7 \times 11$
Thus:
The divisors of $616$ can be enumerated as:
:$1, 2, 4, 7, 8, 11, 14, 22, 28, 44, 56, 77, 88, 154, 308, 616$
Category:Examples of Divisor Count Function
Category:616
Divisor Count of 625

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {625} = 5$
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:
:$625 = 5^4$
Thus:
The divisors of $625$ can be enumerated as:
:$1, 5, 25, 125, 625$
Category:Examples of Divisor Count Function
Category:625
Divisor Count of 630

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {630} = 24$
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:
:$630 = 2 \times 3^2 \times 5 \times 7$
Thus:
The divisors of $630$ can be enumerated as:
:$1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 30, 35, 42, 45, 63, 70, 90, 105, 126, 210, 315, 630$
Category:Examples of Divisor Count Function
Category:630
Divisor Count of 64

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {64} = 7$
where $\sigma_0$ denotes the divisor count function.
== Proof ==
The divisors of $64$ can be enumerated as:
:$1, 2, 4, 8, 16, 32, 64$
Category:Examples of Divisor Count Function
Category:64
Divisor Count of 64,000

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {64 \, 000} = 40$
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:
:$64 \, 000 = 2^9 \times 5^3$
Thus:
The divisors of $64 \, 000$ can be enumerated as:
:$1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100,$
:$125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 640, 800, 1000, 1280,$
:$1600, 2000, 2560, 3200, 4000, 6400, 8000, 12800, 16000, 32000, 64000$
Category:Examples of Divisor Count Function
Category:64,000
Divisor Count of 650

== Example of Use of Divisor Count Function ==
:$\map {\sigma_0} {650} = 12$
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:
:$650 = 2 \times 5^2 \times 13$
Thus:
The divisors of $650$ can be enumerated as:
:$1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650$
Category:Examples of Divisor Count Function
Category:650