nyuuzyou/wikis huggingface dataset:
Divisor Sum of 288

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {288} = 819$
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:
:$288 = 2^5 \times 3^2$
Hence:
Category:Divisor Sum of Integer
Category:288
Divisor Sum of 29,295

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {29 \, 295} = 61 \, 440$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$29 \, 295 = 3^3 \times 5 \times 7 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:29,295
Divisor Sum of 29,912,035,725

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {29 \, 912 \, 035 \, 725} = 64 \, 795 \, 852 \, 800$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$29 \, 912 \, 035 \, 725 = 3^3 \times 5^2 \times 11 \times 13 \times 431 \times 719$
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:29,912,035,725
Divisor Sum of 29,925

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {29 \, 925} = 64 \, 480$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$29 \, 925 = 3^2 \times 5^2 \times 7 \times 19$
Hence:
Category:Divisor Sum of Integer
Category:29,925
Divisor Sum of 2924

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2924} = 5544$
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:
:$2924 = 2^2 \times 17 \times 43$
Hence:
Category:Divisor Sum of Integer
Category:2924
Divisor Sum of 294,896

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {294 \, 896} = 653 \, 232$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$294 \, 896 = 2^4 \times 7 \times 2633$
Hence:
Category:Divisor Sum of Integer
Category:294,896
Divisor Sum of 295

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {295} = 360$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$295 = 5 \times 59$
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:295
Divisor Sum of 295,488

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {295 \, 488} = 924 \, 560$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$295 \, 488 = 2^6 \times 3^5 \times 19$
Hence:
Category:Divisor Sum of Integer
Category:295,488
Divisor Sum of 3

== Example of Divisor Sum of Prime Number ==
:$\map {\sigma_1} 3 = 4$
where ${\sigma_1}$ denotes the divisor sum.
== Proof ==
From Divisor Sum of Prime Number:
:$\map {\sigma_1} p = p + 1$
where $p$ is a prime number.
We have that $3$ is a prime number.
Hence:
:$\map {\sigma_1} 3 = 3 + 1 = 4$
== Sources ==
* 
Category:Examples of Divisor Sum Function
Category:Divisor Sum of Prime Number
Category:Divisor Sum of Integer
Category:3
Divisor Sum of 30

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {30} = 72$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$30 = 2 \times 3 \times 5$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:30
Divisor Sum of 30,240

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {30 \, 240} = 120 \, 960$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$30 \, 240 = 2^5 \times 3^3 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:30,240
Divisor Sum of 30,555

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {30 \, 555} = 61 \, 152$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$30 \, 555 = 3^2 \times 5 \times 7 \times 97$
Hence:
Category:Divisor Sum of Integer
Category:30,555
Divisor Sum of 300

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {300} = 868$
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:
:$300 = 2^2 \times 3 \times 5^2$
Hence:
Category:Divisor Sum of Integer
Category:300
Divisor Sum of 31,185

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {31 \, 185} = 69 \, 696$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$31 \, 185 = 3^4 \times 5 \times 7 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:31,185
Divisor Sum of 31,695,652,275

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {31 \, 695 \, 652 \, 275} = 64 \, 795 \, 852 \, 800$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$31 \, 695 \, 652 \, 275 = 3 \times 5^2 \times 7 \times 19 \times 53 \times 167 \times 359$
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:31,695,652,275
Divisor Sum of 31,704

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {31 \, 704} = 79 \, 320$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$31 \, 704 = 2^3 \times 3 \times 1321$
Hence:
Category:Divisor Sum of Integer
Category:31,704
Divisor Sum of 31,815

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {31 \, 815} = 63 \, 648$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$31 \, 815 = 3^2 \times 5 \times 7 \times 101$
Hence:
Category:Divisor Sum of Integer
Category:31,815
Divisor Sum of 310

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {310} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$310 = 2 \times 5 \times 31$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:310
Divisor Sum of 32,129,958,525

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {32 \, 129 \, 958 \, 525} = 64 \, 795 \, 852 \, 800$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$32 \, 129 \, 958 \, 525 = 3 \times 5^2 \times 7 \times 31 \times 41 \times 179 \times 269$
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:32,129,958,525
Divisor Sum of 32,400

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {32 \, 400} = 116 \, 281$
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:
:$32 \, 400 = 2^4 \times 3^4 \times 5^2$
Hence:
Category:Divisor Sum of Integer
Category:32,400
Category:Numbers whose Divisor Sum is Square
Divisor Sum of 32,445

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {32 \, 445} = 64 \, 896$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$32 \, 445 = 3^2 \times 5 \times 7 \times 103$
Hence:
Category:Divisor Sum of Integer
Category:32,445
Divisor Sum of 32,665,894,275

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {32 \, 665 \, 894 \, 275} = 64 \, 795 \, 852 \, 800$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$32 \, 665 \, 894 \, 275 = 3 \times 5^2 \times 7 \times 31 \times 59 \times 34 \, 019$
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:32,665,894,275
Divisor Sum of 322

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {322} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$322 = 2 \times 7 \times 23$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:322
Divisor Sum of 323

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {323} = 360$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$323 = 17 \times 19$
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:323
Divisor Sum of 33,075

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {33 \, 075} = 70 \, 680$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$33 \, 075 = 3^3 \times 5^2 \times 7^2$
Hence:
Category:Divisor Sum of Integer
Category:33,075
Divisor Sum of 33,100,200,525

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {33 \, 100 \, 200 \, 525} = 64 \, 795 \, 852 \, 800$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$33 \, 100 \, 200 \, 525 = 3 \times 5^2 \times 7 \times 29 \times 971 \times 2239$
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:33,100,200,525
Divisor Sum of 33,705

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {33 \, 705} = 67 \, 392$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$33 \, 705 = 3^2 \times 5 \times 7 \times 107$
Hence:
Category:Divisor Sum of Integer
Category:33,705
Divisor Sum of 33,817,088

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {33 \, 817 \, 088} = 67 \, 832 \, 061$
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:
:$33 \, 817 \, 088 = 2^9 \times 257^2$
Hence:
Category:Divisor Sum of Integer
Category:33,817,088
Divisor Sum of 336

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {336} = 992$
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:
:$336 = 2^4 \times 3 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:336
Divisor Sum of 34,883,817,075

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {34 \, 883 \, 817 \, 075} = 64 \, 795 \, 852 \, 800$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$34 \, 883 \, 817 \, 075 = 3^3 \times 5^2 \times 127 \times 503 \times 809$
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:34,883,817,075
Divisor Sum of 343

== Example of Divisor Sum of Power of Prime ==
:$\map {\sigma_1} {343} = 400$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Power of Prime:
:$\map {\sigma_1} {p^k} = \dfrac {p^{k + 1} - 1} {p_i - 1}$
We have that:
:$343 = 7^3$
Hence:
Thus we have that:
:$7^0 + 7^2 + 7^2 + 7^3 = 20^2$
== Sources ==
* : $400$
* : $400$
Category:Divisor Sum of Power of Prime
Category:Divisor Sum of Integer
Category:343
Category:400
Divisor Sum of 345

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {345} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$345 = 3 \times 5 \times 23$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:345
Divisor Sum of 3465

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {3465} = 7488$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$3465 = 3^2 \times 5 \times 7 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:3465
Divisor Sum of 357

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {357} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$357 = 3 \times 7 \times 17$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:357
Divisor Sum of 358,336

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {358 \, 336} = 777 \, 240$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$358 \, 336 = 2^6 \times 11 \times 509$
Hence:
Category:Divisor Sum of Integer
Category:358,336
Divisor Sum of 360

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {360} = 1170$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$360 = 2^3 \times 3^2 \times 5$
Hence:
Category:Divisor Sum of Integer
Category:360
Divisor Sum of 362

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {362} = 546$
== Proof ==
We have that:
:$362 = 2 \times 181$
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:362
Divisor Sum of 364

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {364} = 784$
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:
:$364 = 2^2 \times 7 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:364
Divisor Sum of 366

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {366} = 744$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$366 = 2 \times 3 \times 61$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:366
Divisor Sum of 366,556

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {366 \, 556} = 641 \, 480$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$366 \, 556 = 2^2 \times 91 \, 639$
Hence:
Category:Divisor Sum of Integer
Category:366,556
Divisor Sum of 368

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {368} = 744$
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:
:$368 = 2^4 \times 23$
Hence:
Category:Divisor Sum of Integer
Category:368
Divisor Sum of 376,736

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {376 \, 736} = 757 \, 764$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$376 \, 736 = 2^5 \times 61 \times 193$
Hence:
Category:Divisor Sum of Integer
Category:376,736
Divisor Sum of 381,028

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {381 \, 028} = 666 \, 806$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$381 \, 028 = 2^2 \times 95 \, 257$
Hence:
Category:Divisor Sum of Integer
Category:381,028
Divisor Sum of 382

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {382} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$382 = 2 \times 191$
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:382
Divisor Sum of 385

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {385} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$385 = 5 \times 7 \times 11$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:385
Divisor Sum of 400

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {400} = 961$
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:
:$400 = 2^4 \times 5^2$
Hence:
Category:Divisor Sum of Integer
Category:400
Divisor Sum of 4030

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {4030} = 8064$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$4030 = 2 \times 5 \times 13 \times 31$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:4030
Divisor Sum of 4095

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {4095} = 8736$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$4095 = 3^2 \times 5 \times 7 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:4095
Divisor Sum of 418

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {418} = 720$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$418 = 2 \times 11 \times 19$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:418
Divisor Sum of 418,904

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {418 \, 904} = 785 \, 460$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$418 \, 904 = 2^3 \times 52 \, 363$
Hence:
Category:Divisor Sum of Integer
Category:418,904
Divisor Sum of 42

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {42} = 96$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$42 = 2 \times 3 \times 7$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:42
Divisor Sum of 420

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {420} = 1344$
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:
:$420 = 2^2 \times 3 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:420
Divisor Sum of 45,045

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {45 \, 045} = 104 \, 832$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$45 \, 045 = 3^2 \times 5 \times 7 \times 11 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:45,045
Divisor Sum of 45,946

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {45 \, 946} = 68 \, 922$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$45 \, 946 = 2 \times 22 \, 973$
Hence:
Category:Divisor Sum of Integer
Category:45,946
Divisor Sum of 4536

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {4536} = 14 \, 520$
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:
:$4536 = 2^3 \times 3^4 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:4536
Divisor Sum of 459,818,240

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {459 \, 818 \, 240} = 1 \, 379 \, 454 \, 720$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$459 \, 818 \, 240 = 2^8 \times 5 \times 7 \times 19 \times 37 \times 73$
Hence:
Category:Divisor Sum of Integer
Category:459,818,240
Divisor Sum of 46

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {46} = 72$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$46 = 2 \times 23$
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:46
Divisor Sum of 47,616

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {47 \, 616} = 130 \, 994$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$47 \, 616 = 2^9 \times 3 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:47,616
Divisor Sum of 472

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {472} = 900$
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:
:$472 = 2^3 \times 59$
Hence:
Category:Divisor Sum of Integer
Category:472
Divisor Sum of 4725

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {4725} = 9920$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$4725 = 3^3 \times 5^2 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:4725
Divisor Sum of 48,976

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {48 \, 976} = 94 \, 922$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$48 \, 976 = 2^4 \times 3061$
Hence:
Category:Divisor Sum of Integer
Category:48,976
Divisor Sum of 480

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {480} = 1512$
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:
:$480 = 2^5 \times 3 \times 5$
Hence:
Category:Divisor Sum of Integer
Category:480
Divisor Sum of 496

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {496} = 992$
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:
:$496 = 2^4 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:496
Divisor Sum of 497

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {497} = 576$
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:
:$497 = 7 \times 71$
Hence:
Category:Divisor Sum of Integer
Category:497
Divisor Sum of 5,391,411,025

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {5 \, 391 \, 411 \, 025} = 10 \, 799 \, 308 \, 800$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$5 \, 391 \, 411 \, 025 = 5^2 \times 7 \times 11 \times 13 \times 17 \times 19 \times 23 \times 29$
Hence:
Category:Divisor Sum of Integer
Category:5,391,411,025
Divisor Sum of 5002

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {5002} = 7812$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$5002 = 2 \times 41 \times 61$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:5002
Divisor Sum of 5020

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {5020} = 10 \, 584$
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 $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$5020 = 2^2 \times 5 \times 251$
Hence:
Category:Divisor Sum of Integer
Category:5020