nyuuzyou/wikis huggingface dataset:
Divisor Sum of 15,472

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {15 \, 472} = 30 \, 008$
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:
:$15 \, 472 = 2^4 \times 967$
Hence:
Category:Divisor Sum of Integer
Category:15,472
Divisor Sum of 152,990

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {152 \, 990} = 275 \, 400$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$152 \, 990 = 2 \times 5 \times 15 \, 299$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:152,990
Divisor Sum of 154,345,556,085,770,649,600

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600} = 926 \, 073 \, 336 \, 514 \, 623 \, 897 \, 600$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600 = 2^{15} \times 3^5 \times 5^2 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 31 \times 43 \times 257$
Hence:
Category:Divisor Sum of Integer
Category:154,345,556,085,770,649,600
Divisor Sum of 1575

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1575} = 3224$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$1575 = 3^2 \times 5^2 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:1575
Divisor Sum of 16,065

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {16 \, 065} = 34 \, 560$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$16 \, 065 = 3^3 \times 5 \times 7 \times 17$
Hence:
Category:Divisor Sum of Integer
Category:16,065
Divisor Sum of 16,695

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {16 \, 695} = 33 \, 696$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$16 \, 695 = 3^2 \times 5 \times 7 \times 53$
Hence:
Category:Divisor Sum of Integer
Category:16,695
Divisor Sum of 1638

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1638} = 4368$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$1638 = 2 \times 3^2 \times 7 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:1638
Divisor Sum of 168

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {168} = 480$
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:
:$168 = 2^3 \times 3 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:168
Divisor Sum of 1680

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1680} = 5952$
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:
:$1680 = 2^4 \times 3 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:1680
Divisor Sum of 17,296

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {17 \, 296} = 35 \, 712$
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:
:$17 \, 296 = 2^4 \times 23 \times 47$
Hence:
Category:Divisor Sum of Integer
Category:17,296
Divisor Sum of 17,325

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {17 \, 325} = 38 \, 688$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$17 \, 325 = 3^2 \times 5^2 \times 7 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:17,325
Divisor Sum of 17,716

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {17 \, 716} = 32 \, 032$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$17 \, 716 = 2^2 \times 43 \times 103$
Hence:
Category:Divisor Sum of Integer
Category:17,716
Divisor Sum of 17,955

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {17 \, 955} = 38 \, 400$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$17 \, 955 = 3^3 \times 5 \times 7 \times 19$
Hence:
Category:Divisor Sum of Integer
Category:17,955
Divisor Sum of 177,792

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {177 \, 792} = 473 \, 280$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$177 \, 792 = 2^7 \times 3 \times 7 \times 463$
Hence:
Category:Divisor Sum of Integer
Category:177,792
Divisor Sum of 1798

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {1798} = 2880$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$1798 = 2 \times 29 \times 31$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:1798
Divisor Sum of 18,416

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {18 \, 416} = 35 \, 712$
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:
:$18 \, 416 = 2^4 \times 1151$
Hence:
Category:Divisor Sum of Integer
Category:18,416
Divisor Sum of 18,585

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {18 \, 585} = 33 \, 696$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$18 \, 585 = 3^2 \times 5 \times 7 \times 59$
Hence:
Category:Divisor Sum of Integer
Category:18,585
Divisor Sum of 180

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {180} = 546$
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:
:$180 = 2^2 \times 3^2 \times 5$
Hence:
Category:Divisor Sum of Integer
Category:180
Divisor Sum of 19,116

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {19 \, 116} = 50 \, 820$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$19 \, 116 = 2^2 \times 3^4 \times 59$
Hence:
Category:Divisor Sum of Integer
Category:19,116
Divisor Sum of 19,215

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {19 \, 215} = 38 \, 688$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$19 \, 215 = 3^2 \times 5 \times 7 \times 61$
Hence:
Category:Divisor Sum of Integer
Category:19,215
Divisor Sum of 19,845

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {19 \, 845} = 41 \, 382$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$19 \, 845 = 3^4 \times 5 \times 7^2$
Hence:
Category:Divisor Sum of Integer
Category:19,845
Divisor Sum of 19,916

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {19 \, 916} = 37 \, 632$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$19 \, 916 = 2^2 \times 13 \times 383$
Hence:
Category:Divisor Sum of Integer
Category:19,916
Divisor Sum of 190

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {190} = 360$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$190 = 2 \times 5 \times 19$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:190
Divisor Sum of 1925

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1925} = 2976$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From Divisor Sum of Integer: Corollary
:$\ds \map {\sigma_1} n = \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop > 1} } \frac {p_i^{k_i + 1} - 1} {p_i - 1} \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop = 1} } \paren {p_i + 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$1925 = 5^2 \times 7 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:1925
Divisor Sum of 195

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {195} = 336$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$195 = 3 \times 5 \times 13$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:195
Divisor Sum of 1980

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1980} = 6552$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$1980 = 2^2 \times 3^2 \times 5 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:1980
Divisor Sum of 2

== Example of Divisor Sum of Prime Number ==
:$\map {\sigma_1} 2 = 3$
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 $2$ is a prime number.
Hence:
:$\map {\sigma_1} 2 = 2 + 1 = 3$
Category:Examples of Divisor Sum Function
Category:Divisor Sum of Prime Number
Category:2
Divisor Sum of 20,475

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {20 \, 475} = 45 \, 136$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$20 \, 475 = 3^2 \times 5^2 \times 7 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:20,475
Divisor Sum of 2016

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2016} = 6552$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$2016 = 2^5 \times 3^2 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:2016
Divisor Sum of 2024

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2024} = 4320$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$2024 = 2^3 \times 11 \times 23$
Hence:
Category:Divisor Sum of Integer
Category:2024
Divisor Sum of 2025

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2025} = 7502$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$2025 = 3^4 \times 5^2$
Hence:
Category:Divisor Sum of Integer
Category:2025
Divisor Sum of 21,105

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {21 \, 105} = 42 \, 432$
where $\sigma_1$ denotes the Divisor sum function.
== Proof ==
We have that:
:$21 \, 105 = 3^2 \times 5 \times 7 \times 67$
Hence:
Category:Divisor Sum of Integer
Category:21,105
Divisor Sum of 21,735

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {21 \, 735} = 46 \, 080$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$21 \, 735 = 3^3 \times 5 \times 7 \times 23$
Hence:
Category:Divisor Sum of Integer
Category:21,735
Divisor Sum of 210

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {210} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$210 = 2 \times 3 \times 5 \times 7$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:210
Divisor Sum of 214

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {214} = 324$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$214 = 2 \times 107$
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:214
Divisor Sum of 217

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {217} = 256$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$217 = 7 \times 31$
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:217
Divisor Sum of 22,365

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {22 \, 365} = 44 \, 928$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$22 \, 365 = 3^2 \times 5 \times 7 \times 71$
Hence:
Category:Divisor Sum of Integer
Category:22,365
Divisor Sum of 22,744

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {22 \, 744} = 42 \, 660$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$22 \, 744 = 2^3 \times 2843$
Hence:
Category:Divisor Sum of Integer
Category:22,744
Divisor Sum of 22,976

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {22 \, 976} = 45 \, 720$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$22 \, 976 = 2^6 \times 359$
Hence:
Category:Divisor Sum of Integer
Category:22,976
Divisor Sum of 22,995

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {22 \, 995} = 46 \, 176$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$22 \, 995 = 3^2 \times 5 \times 7 \times 73$
Hence:
Category:Divisor Sum of Integer
Category:22,995
Divisor Sum of 220

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {220} = 504$
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:
:$220 = 2^2 \times 5 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:220
Divisor Sum of 2205

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2205} = 4446$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$2205 = 3^2 \times 5 \times 7^2$
Hence:
Category:Divisor Sum of Integer
Category:2205
Divisor Sum of 23,625

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {23 \, 625} = 49 \, 920$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$23 \, 625 = 3^3 \times 5^3 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:23,625
Divisor Sum of 24,255

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {24 \, 255} = 53 \, 352$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$24 \, 255 = 3^2 \times 5 \times 7^2 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:24,255
Divisor Sum of 24,885

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {24 \, 885} = 49 \, 920$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$24 \, 885 = 3^2 \times 5 \times 7 \times 79$
Hence:
Category:Divisor Sum of Integer
Category:24,885
Divisor Sum of 240

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {240} = 744$
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:
:$240 = 2^4 \times 3 \times 5$
Hence:
Category:Divisor Sum of Integer
Category:240
Divisor Sum of 2400

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2400} = 7812$
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:
:$2400 = 2^5 \times 3 \times 5^2$
Hence:
Category:Divisor Sum of Integer
Category:2400
Divisor Sum of 243,760

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {243 \, 760} = 620 \, 496$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$243 \, 760 = 2^4 \times 5 \times 11 \times 277$
Hence:
Category:Divisor Sum of Integer
Category:243,760
Divisor Sum of 248

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {248} = 480$
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:
:$248 = 2^3 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:248
Divisor Sum of 25,515

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {25 \, 515} = 52 \, 464$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$25 \, 515 = 3^6 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:25,515
Divisor Sum of 2520

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2520} = 9360$
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:
:$2520 = 2^3 \times 3^2 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:2520
Divisor Sum of 2556

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2556} = 6552$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$2556 = 2^2 \times 3^2 \times 71$
Hence:
Category:Divisor Sum of Integer
Category:2556
Divisor Sum of 26,145

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {26 \, 145} = 52 \, 416$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$26 \, 145 = 3^2 \times 5 \times 7 \times 83$
Hence:
Category:Divisor Sum of Integer
Category:26,145
Divisor Sum of 26,775

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {26 \, 775} = 58 \, 032$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$26 \, 775 = 3^2 \times 5^2 \times 7 \times 17$
Hence:
Category:Divisor Sum of Integer
Category:26,775
Divisor Sum of 2620

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2620} = 5544$
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:
:$2620 = 2^2 \times 5 \times 131$
Hence:
Category:Divisor Sum of Integer
Category:2620
Divisor Sum of 264

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {264} = 720$
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:
:$264 = 2^3 \times 3 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:264
Divisor Sum of 265

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {265} = 324$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$265 = 5 \times 53$
Hence:
Category:Divisor Sum of Non-Square Semiprime
Category:Divisor Sum of Integer
Category:265
Divisor Sum of 267

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {267} = 360$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$267 = 3 \times 89$
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:267
Divisor Sum of 27,405

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {27 \, 405} = 57 \, 600$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$27 \, 405 = 3^3 \times 5 \times 7 \times 29$
Hence:
Category:Divisor Sum of Integer
Category:27,405
Divisor Sum of 27,418, 521,963, 671,501, 273,905, 190,135, 082,692, 041,730, 405,303, 870,249, 023,209

== 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:27,418, 521,963, 671,501, 273,905, 190,135, 082,692, 041,730, 405,303, 870,249, 023,209
Divisor Sum of 270

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {270} = 720$
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:
:$270 = 2 \times 3^3 \times 5$
Hence:
Category:Divisor Sum of Integer
Category:270
Divisor Sum of 274,924

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {274 \, 924} = 550 \, 368$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$274 \, 924 = 2^2 \times 13 \times 17 \times 311$
Hence:
Category:Divisor Sum of Integer
Category:274,924
Divisor Sum of 275,444

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {275 \, 444} = 519 \, 204$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$275 \, 444 = 2^2 \times 13 \times 5297$
Hence:
Category:Divisor Sum of Integer
Category:275,444
Divisor Sum of 28

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {28} = 56$
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:
:$28 = 2^2 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:28
Divisor Sum of 28,035

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {28 \, 035} = 56 \, 160$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$28 \, 035 = 3^2 \times 5 \times 7 \times 89$
Hence:
Category:Divisor Sum of Integer
Category:28,035
Divisor Sum of 28,665

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {28 \, 665} = 62 \, 244$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$28 \, 665 = 3^2 \times 5 \times 7^2 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:28,665
Divisor Sum of 282

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {282} = 576$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$282 = 2 \times 3 \times 47$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:282
Divisor Sum of 2835

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {2835} = 5808$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$2835 = 3^4 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:2835
Divisor Sum of 284

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {284} = 504$
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:
:$284 = 2^2 \times 71$
Hence:
Category:Divisor Sum of Integer
Category:284
Divisor Sum of 285,778

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {285 \, 778} = 438 \, 768$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$285 \, 778 = 2 \times 43 \times 3323$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:285,778
Divisor Sum of 286

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {286} = 504$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$286 = 2 \times 11 \times 13$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:286