nyuuzyou/wikis huggingface dataset:
Divisor Sum of 504

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {504} = 1560$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$504 = 2^3 \times 3^2 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:504
Divisor Sum of 5040

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {5040} = 19 \, 344$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$5040 = 2^4 \times 3^2 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:5040
Divisor Sum of 51,001,180,160

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {51 \, 001 \, 180 \, 160} = 153 \, 003 \, 540 \, 480$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$51 \, 001 \, 180 \, 160 = 2^{14} \times 5 \times 7 \times 19 \times 31 \times 151$
Hence:
Category:Divisor Sum of Integer
Category:51,001,180,160
Divisor Sum of 510

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {510} = 1296$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer: Corollary
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$510 = 2 \times 3 \times 5 \times 17$
Hence:
Category:Divisor Sum of Integer
Category:510
Divisor Sum of 517

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {517} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer: Corollary
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$517 = 11 \times 47$
Hence:
Category:Divisor Sum of Integer
Category:517
Divisor Sum of 523,776

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {523 \, 776} = 1 \, 571 \, 328$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$523 \, 776 = 2^9 \times 3 \times 11 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:523,776
Divisor Sum of 527

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {527} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer: Corollary
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$527 = 17 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:527
Divisor Sum of 5355

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {5355} = 11 \, 232$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$5355 = 3^2 \times 5 \times 7 \times 17$
Hence:
Category:Divisor Sum of Integer
Category:5355
Divisor Sum of 5564

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {5564} = 10 \, 584$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$5564 = 2^2 \times 13 \times 107$
Hence:
Category:Divisor Sum of Integer
Category:5564
Divisor Sum of 570

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {570} = 1440$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$570 = 2 \times 3 \times 5 \times 19$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:570
Divisor Sum of 5775

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {5775} = 11 \, 904$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$5775 = 3 \times 5^2 \times 7 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:5775
Divisor Sum of 5830

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {5830} = 11 \, 664$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$5830 = 2 \times 5 \times 11 \times 53$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:5830
Divisor Sum of 589,786

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {589 \, 786} = 884 \, 682$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$589 \, 786 = 2 \times 294 \, 893$
Hence:
Category:Divisor Sum of Non-Square Semiprime
Category:Divisor Sum of Integer
Category:589,786
Divisor Sum of 59,554, 936,495, 441,481, 044,788, 091,271, 148,664, 944,796, 300,859, 243,635, 311,219, 048,448

== Example of Divisor Sum of Integer ==
where $\sigma_1$ denotes the divisor sum.
== Proof ==
We have that:
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:59,554, 936,495, 441,481, 044,788, 091,271, 148,664, 944,796, 300,859, 243,635, 311,219, 048,448
Divisor Sum of 59,554, 936,495, 441,891, 385,123, 332,422, 108,719, 776,971, 992,921, 810,832, 072,976, 105,472

== Example of Divisor Sum of Integer ==
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:59,554, 936,495, 441,891, 385,123, 332,422, 108,719, 776,971, 992,921, 810,832, 072,976, 105,472
Divisor Sum of 5985

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {5985} = 12 \, 480$
where $\sigma_1$ denotes the divsor sum function.
== Proof ==
We have that:
:$5985 = 3^2 \times 5 \times 7 \times 19$
Hence:
Category:Divisor Sum of Integer
Category:5985
Divisor Sum of 6

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} 6 = 12$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$6 = 2 \times 3$
Hence:
== Sources ==
* 
Category:Divisor Sum of Non-Square Semiprime
Category:Divisor Sum of Integer
Category:6
Divisor Sum of 600

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {600} = 1860$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$600 = 2^3 \times 3 \times 5^2$
Hence:
Category:Divisor Sum of Integer
Category:600
Divisor Sum of 616

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {616} = 1440$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$616 = 2^3 \times 7 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:616
Divisor Sum of 62

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {62} = 96$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$62 = 2 \times 31$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
Category:Divisor Sum of Non-Square Semiprime
Category:Divisor Sum of Integer
Category:62
Divisor Sum of 6232

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {6232} = 12 \, 600$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$6232 = 2^3 \times 19 \times 41$
Hence:
Category:Divisor Sum of Integer
Category:6232
Divisor Sum of 629,072

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {629 \, 072} = 1 \, 218 \, 858$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$629 \, 072 = 2^4 \times 39 \, 317$
Hence:
Category:Divisor Sum of Integer
Category:629,072
Divisor Sum of 630

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {630} = 1872$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$630 = 2 \times 3^2 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:630
Divisor Sum of 6368

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {6368} = 12 \, 600$
where $\sigma_1$ denotes the Divisor sum function.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$6368 = 2^5 \times 199$
Hence:
Category:Divisor Sum of Integer
Category:6368
Divisor Sum of 6435

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {6435} = 13 \, 104$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$6435 = 3^2 \times 5 \times 11 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:6435
Divisor Sum of 65

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {65} = 84$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$65 = 5 \times 13$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
Category:Divisor Sum of Non-Square Semiprime
Category:Divisor Sum of Integer
Category:65
Divisor Sum of 660

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {660} = 2016$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$660 = 2^2 \times 3 \times 5 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:660
Divisor Sum of 6615

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {6615} = 13 \, 680$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$6615 = 3^3 \times 5 \times 7^2$
Hence:
Category:Divisor Sum of Integer
Category:6615
Divisor Sum of 6825

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {6825} = 13 \, 888$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$6825 = 3 \times 5^2 \times 7 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:6825
Divisor Sum of 714

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {714} = 1728$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$714 = 2 \times 3 \times 7 \times 17$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:714
Divisor Sum of 7192

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {7192} = 14 \, 400$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$7192 = 2^3 \times 29 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:7192
Divisor Sum of 720

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {720} = 2418$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$720 = 2^4 \times 3^2 \times 5$
Hence:
Category:Divisor Sum of Integer
Category:720
Divisor Sum of 7245

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {7245} = 14 \, 976$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$7245 = 3^2 \times 5 \times 7 \times 23$
Hence:
Category:Divisor Sum of Integer
Category:7245
Divisor Sum of 74

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {74} = 114$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$74 = 2 \times 37$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
Category:Divisor Sum of Non-Square Semiprime
Category:Divisor Sum of Integer
Category:74
Divisor Sum of 7425

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {7425} = 14 \, 880$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$7425 = 3^3 \times 5^2 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:7425
Divisor Sum of 75

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {75} = 124$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$75 = 3 \times 5^2$
Hence:
Category:Divisor Sum of Integer
Category:75
Divisor Sum of 7560

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {7560} = 28 \, 800$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$7560 = 2^3 \times 3^3 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:7560
Divisor Sum of 76

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {76} = 140$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$76 = 2^2 \times 19$
Hence:
Category:Divisor Sum of Integer
Category:76
Divisor Sum of 770

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {770} = 1728$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$770 = 2 \times 5 \times 7 \times 11$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:770
Divisor Sum of 78

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {78} = 168$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$78 = 2 \times 3 \times 13$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:78
Divisor Sum of 7875

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {7875} = 16 \, 224$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$7875 = 3^2 \times 5^3 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:7875
Divisor Sum of 7912

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {7912} = 15 \, 840$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$7912 = 2^3 \times 23 \times 43$
Hence:
Category:Divisor Sum of Integer
Category:7912
Divisor Sum of 8,212,890,625

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {8 \, 212 \, 890 \, 625} = 10 \, 632 \, 324 \, 001$
where $\sigma_1$ denotes the divisor sum.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{i \mathop = 1}^r \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop = 1}^r p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$8 \, 212 \, 890 \, 625 = 5^{10} \times 29^2$
Hence:
Category:Divisor Sum of Integer
Category:8,212,890,625
Divisor Sum of 8085

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {8085} = 16 \, 416$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$8085 = 3 \times 5 \times 7^2 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:8085
Divisor Sum of 81,081

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {81 \, 081} = 162 \, 624$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$81 \, 081 = 3^4 \times 7 \times 11 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:81,081
Divisor Sum of 83,328

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {83 \, 328} = 261 \, 120$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$83 \, 328 = 2^7 \times 3 \times 7 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:83,328
Divisor Sum of 840

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {840} = 2880$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$840 = 2^3 \times 3 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:840
Divisor Sum of 8415

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {8415} = 16 \, 848$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$8415 = 3^2 \times 5 \times 11 \times 17$
Hence:
Category:Divisor Sum of Integer
Category:8415
Divisor Sum of 8505

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {8505} = 17 \, 472$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$8505 = 3^5 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:8505
Divisor Sum of 87

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {87} = 120$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$87 = 3 \times 29$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
Category:Divisor Sum of Non-Square Semiprime
Category:Divisor Sum of Integer
Category:87
Divisor Sum of 9135

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {9135} = 18 \, 720$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$9135 = 3^2 \times 5 \times 7 \times 29$
Hence:
Category:Divisor Sum of Integer
Category:9135
Divisor Sum of 9272

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {9272} = 18 \, 600$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$9272 = 2^3 \times 19 \times 61$
Hence:
Category:Divisor Sum of Integer
Category:9272
Divisor Sum of 9374

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {9374} = 14 \, 520$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$9374 = 2 \times 43 \times 109$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:9374
Divisor Sum of 960

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {960} = 3048$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$960 = 2^6 \times 3 \times 5$
Hence:
Category:Divisor Sum of Integer
Category:960
Divisor Sum of 97,946

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {97 \, 946} = 146 \, 922$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$97 \, 946 = 2 \times 48 \, 973$
Hence:
Category:Divisor Sum of Non-Square Semiprime
Category:97,946
Divisor Sum of 9765

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {9765} = 19 \, 968$
where $\sigma_1$ denotes the divisor dum function.
== Proof ==
We have that:
:$9765 = 3^2 \times 5 \times 7 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:9765
Divisor Sum of 994

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {994} = 1728$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$994 = 2 \times 7 \times 71$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:994
Divisor Sum of Integer

== Theorem ==
Let $n$ be an integer such that $n \ge 2$.
Let $\map {\sigma_1} n$ be the divisor sum of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
Then:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
=== Corollary ===
== Proof ==
We have that the Divisor Sum Function is Multiplicative.
From Value of Multiplicative Function is Product of Values of Prime Power Factors, we have:
:$\map f n = \map f {p_1^{k_1} } \map f {p_2^{k_2} } \ldots \map f {p_r^{k_r} }$
From Divisor Sum of Power of Prime, we have:
:$\ds \map {\sigma_1} {p_i^{k_i} } = \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
Hence the result.
== Examples ==
== Sources ==
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Category:Divisor Sum of Integer
Category:Divisor Sum Function
Divisor Sum of Integer/Corollary

== Corollary to Divisor Sum of Integer ==
Let $n$ be an integer such that $n \ge 2$.
Let the prime decomposition of $n$ be:
:$\ds n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$
Let $\map {\sigma_1} n$ be the divisor sum of $n$.
That is, let $\map {\sigma_1} n$ be the sum of all positive divisors of $n$.
Then:
:$\ds \map {\sigma_1} n = \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop > 1} } \frac {p_i^{k_i + 1} - 1} {p_i - 1} \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop = 1} } \paren {p_i + 1}$
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
Suppose $k_i = 1$.
Then we have:
Thus the contribution from a prime factor which is square-free can be expressed in the simpler form $p_i + 1$ instead of the more unwieldy form $\dfrac {p_i^2 - 1} {p_i - 1}$.
Hence the result.
== Examples ==
Category:Divisor Sum of Integer
Divisor Sum of Integer/Examples

== Examples of Divisor Sum of Integer ==
Let ${\sigma_1}: \Z_{>0} \to \Z_{>0}$ denote the divisor sum function: the sum of all the positive integer divisors of $n$.
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Category:Divisor Sum of Integer
Divisor Sum of Integer/Examples/20

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {20} = 42$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer:
:$\ds \map {\sigma_1} n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$20 = 2^2 \times 5$
Hence:
Category:Divisor Sum of Integer
Category:20