nyuuzyou/wikis huggingface dataset:
Divisor Relation is Primitive Recursive

== Theorem ==
The divisor relation $m \divides n$ in $\N^2$ is primitive recursive.
== Proof ==
We note that $m \divides n \iff n = q m$ where $q \in \Z$.
So we see that $m \divides n \iff \map \rem {n, m} = 0$ (see Remainder is Primitive Recursive).
Thus we define the function $\operatorname{div}: \N^2 \to \N$ as:
:$\map {\operatorname {div} } {n, m} = \map {\chi_{\operatorname {eq} } } {\map \rem {n, m}, 0}$
where $\map {\chi_{\operatorname {eq} } } {n, m}$ denotes the characteristic function of the equality relation.
So we have:
:$\map {\operatorname {div} } {n, y} = \begin{cases}
1 & : y \divides n \\
0 & : y \nmid n
\end{cases}$
So $\map {\operatorname {div} } {n, m}$ is defined by substitution from:
:the primitive recursive function $\rem$
:the primitive recursive relation $\operatorname {eq}$
:the constants $1$ and $0$.
Thus $\operatorname {div}$ is primitive recursive.
Hence the result.
Category:Primitive Recursive Functions
Divisor Relation is Transitive

== Theorem ==
The divisibility relation is a transitive relation on $\Z$, the set of integers.
That is:
:$\forall x, y, z \in \Z: x \divides y \land y \divides z \implies x \divides z$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Introduction $\S 6$: The division process in $I$
* : $\S 11.3$: The division algorithm
* : $\text I$: The Series of Primes: $1.1$ Divisibility of integers
* : Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Theorem $2 \text{-} 2 \ (4)$
* : Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$: $\mathbf D. \, 1$
* : $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization: $\text {(iv)}$
Category:Divisors
Category:Divisor Relation is Transitive
Category:Examples of Transitive Relations
Divisor Relation on Positive Integers is Partial Ordering

== Theorem ==
The divisor relation is a partial ordering of $\Z_{>0}$.
== Proof ==
Checking in turn each of the criteria for an ordering:
=== Divisor Relation is Reflexive ===
=== Divisor Relation is Transitive ===
:$\forall x, y, z \in \Z: x \divides y \land y \divides z \implies x \divides z$:
=== Divisor Relation is Antisymmetric ===
=== Divisor Ordering is Partial ===
Let $a = 2$ and $b = 3$.
Then neither $a \divides b$ nor $b \divides a$.
Thus, while the divisor relation is an ordering, it is specifically a partial ordering
== Sources ==
* : Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Example $14.4$
* : $\S \text I.1$
* : $\S 3.1$: Partially ordered sets: Example $4$
* : $\S 7$: Example $7.2$
* : Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets: $\text{(ii)}$
* : Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$
* : Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: $(2)$
* 
* 
Category:Divisors
Category:Integers
Category:Examples of Orderings
Divisor Relation on Positive Integers is Well-Founded Ordering

== Theorem ==
The divisor relation on $\Z_{>0}$ is a well-founded ordering.
== Proof ==
Let $\struct {\Z_{>0}, \divides}$ denote the relational structure formed from the strictly positive integers $\Z_{>0}$ under the divisor relation $\divides$.
From Divisor Relation on Positive Integers is Partial Ordering, $\struct {\Z_{>0}, \divides}$ is a partially ordered set.
It remains to be shown that $\divides$ is well-founded.
By definition, we need to show:
:For any non-empty set $T \subseteq \Z_{>0}$, there is an element $z \in T$ such that for all $y \in T \setminus \set z$, $y \nmid z$.
We choose $z = \min T$ as per the usual ordering on $\Z_{>0}$.
Then for any $y \in T \setminus \set z$, we have $y > z$.
By the contrapositive of Corollary to Absolute Value of Integer is not less than Divisors, we have $y \nmid z$.
Hence the result.
== Sources ==
* : Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $11$
Category:Divisors
Category:Integers
Category:Examples of Well-Founded Relations
Divisor Sum Function is Multiplicative

== Theorem ==
The divisor sum function:
:$\ds {\sigma_1}: \Z_{>0} \to \Z_{>0}: \map {\sigma_1} n = \sum_{d \mathop \divides n} d$
is multiplicative.
== Proof 1 ==
== Proof 2 ==
Category:Multiplicative Functions
Category:Divisor Sum Function
Category:Divisor Sum Function is Multiplicative
Divisor Sum Function is Multiplicative/Proof 1

== Theorem ==
== Proof ==
Let $I_{\Z_{>0}}: \Z_{>0} \to \Z_{>0}$ be the identity function:
:$\forall n \in \Z_{>0}: \map {I_{\Z_{>0} } } n = n$
Thus we have:
:$\ds \map {\sigma_1} n = \sum_{d \mathop \divides n} d = \sum_{d \mathop \divides n} \map {I_{\Z_{>0} } } d$
But from Identity Function is Completely Multiplicative, $I_{\Z_{>0} }$ is multiplicative.
The result follows from Sum Over Divisors of Multiplicative Function.
Category:Divisor Sum Function is Multiplicative
Divisor Sum Function is Multiplicative/Proof 2

== Theorem ==
== Proof ==
Let $a, b$ be coprime integers.
Because $a$ and $b$ have no common divisor, the divisors of $a b$ are integers of the form $a_i b_j$, where $a_i$ is a divisor of $a$ and $b_j$ is a divisor of $b$.
That is, any divisor $d$ of $a b$ is in the form:
:$d = a_i b_j$
in a unique way, where $a_i \divides a$ and $b_j \divides b$.
We can list the divisors of $a$ and $b$ as 
:$1, a_1, a_2, \ldots, a$
and:
:$1, b_1, b_2, \ldots, b$
and so their divisor sums are:
:$\ds \map {\sigma_1} a = \sum_{i \mathop = 1}^r a_i$
:$\ds \map {\sigma_1} b = \sum_{j \mathop = 1}^s b_j$
Consider all divisors of $a b$ with the same $a_i$.
Their sum is:
Summing for all $a_i$:
== Sources ==
* : Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes
Category:Divisor Sum Function is Multiplicative
Divisor Sum is Odd iff Argument is Square or Twice Square

== Theorem ==
Let $\sigma_1: \Z \to \Z$ be the divisor sum function.
Then $\map {\sigma_1} n$ is odd $n$ is either square or twice a square.
== Proof ==
Let $n$ be an integer such that $n \ge 2$.
Let the prime decomposition of $n$ be:
:$n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$
Then:
Let $\map {\sigma_1} n$ be odd.
Then all factors of $\ds \prod_{i \mathop = 1}^r \paren {1 + p_i + p_i^2 + \ldots + p_i^{k_i} }$ are odd (and of course $\ge 3$).
For $1 + p_i + p_i^2 + \ldots + p_i^{k_i}$ to be odd, one of two conditions must hold:
:$p_i$ is even (so that all terms of $1 + p_i + p_i^2 + \ldots + p_i^{k_i}$ are even except the $1$);
:$k_i$ is even (so that $1 + p_i + p_i^2 + \ldots + p_i^{k_i}$ has an odd number of odd terms).
In the first case, that means $p_i^{k_i}$ is a power of $2$.
In the second case, that means $p_i^{k_i}$ is a square.
The result follows.
The argument reverses.
Category:Square Numbers
Category:Divisor Sum Function
Category:Odd Integers
Divisor Sum of 1

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} 1 = 1$
where $\sigma_1$ denotes the divisor sum.
== Proof ==
By definition, the divisor sum of an integer $n$ is the sum of the positive integer divisors of $n$.
There is only one positive integer which is a divisor of $1$, and that is $1$ itself.
Hence the result.
Category:Divisor Sum of Integer
Divisor Sum of 1,175,265

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1 \, 175 \, 265} = 2 \, 614 \, 248$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$1 \, 175 \, 265 = 3^2 \times 5 \times 7^2 \times 13 \times 41$
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:1,175,265
Divisor Sum of 1,438,983

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1 \, 438 \, 983} = 2 \, 614 \, 248$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$1 \, 438 \, 983 = 3^2 \times 7^2 \times 13 \times 251$
Hence from Divisor Sum of Integer:
Category:Divisor Sum of Integer
Category:1,438,983
Divisor Sum of 1,476,304,896

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1 \, 476 \, 304 \, 896} = 4 \, 428 \, 914 \, 688$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$1 \, 476 \, 304 \, 896 = 2^{13} \times 3 \times 11 \times 43 \times 127$
Hence:
Category:Divisor Sum of Integer
Category:1,476,304,896
Divisor Sum of 1,571,328

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1 \, 571 \, 328} = 5 \, 106 \, 816$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$1 \, 571 \, 328 = 2^9 \times 3^2 \times 11 \times 31$
Hence:
Category:Divisor Sum of Integer
Category:1,571,328
Divisor Sum of 1,705,636

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1 \, 705 \, 636} = 2 \, 989 \, 441$
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:
:$1 \, 705 \, 636 = 2^2 \times 653^2$
Hence:
Category:Divisor Sum of Integer
Category:1,705,636
Category:Numbers whose Divisor Sum is Square
Divisor Sum of 1,857,437,604

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1 \, 857 \, 437 \, 604} = 5 \, 168 \, 743 \, 489$
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:
:$1 \, 857 \, 437 \, 604 = 2^2 \times 3^2 \times 11^2 \times 653^2$
Hence:
Category:Divisor Sum of Integer
Category:1,857,437,604
Divisor Sum of 10,395

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {10 \, 395} = 23 \, 040$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$10 \, 395 = 3^3 \times 5 \times 7 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:10,395
Divisor Sum of 10,744

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {10 \, 744} = 21 \, 600$
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:
:$10 \, 744 = 2^3 \times 17 \times 79$
Hence:
Category:Divisor Sum of Integer
Category:10,744
Divisor Sum of 10,856

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {10 \, 856} = 21 \, 600$
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:
:$10 \, 856 = 2^3 \times 23 \times 59$
Hence:
Category:Divisor Sum of Integer
Category:10,856
Divisor Sum of 1008

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1008} = 3224$
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:
:$1008 = 2^4 \times 3^2 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:1008
Divisor Sum of 102

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {102} = 216$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$102 = 2 \times 3 \times 17$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:102
Divisor Sum of 103,340,640

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {103 \, 340 \, 640} = 350 \, 584 \, 416$
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:
:$103 \, 340 \, 640 = 2^5 \times 3 \times 5 \times 13 \times 16 \, 561$
Hence:
Category:Divisor Sum of Integer
Category:103,340,640
Divisor Sum of 1034

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {1034} = 1728$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$1034 = 2 \times 11 \times 47$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:1034
Divisor Sum of 105

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {105} = 192$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$105 = 3 \times 5 \times 7$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:105
Divisor Sum of 1050

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1050} = 2976$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
From 
:$\ds \map {\sigma_1} n = \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop > 1} } \frac {p_i^{k_i + 1} - 1} {p_i - 1} \prod_{\substack {1 \mathop \le i \mathop \le r \\ k_i \mathop = 1} } \paren {p_i + 1}$
where $n = \ds \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$.
We have that:
:$1050 = 2 \times 3 \times 5^2 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:1050
Divisor Sum of 106

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {106} = 162$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$106 = 2 \times 53$
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:106
Divisor Sum of 1080

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1080} = 3600$
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:
:$1080 = 2^3 \times 3^3 \times 5$
Hence:
Category:Divisor Sum of Integer
Category:1080
Divisor Sum of 11,025

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {11 \, 025} = 22 \, 971$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$11 \, 025 = 3^2 \times 5^2 \times 7^2$
Hence:
Category:Divisor Sum of Integer
Category:11,025
Divisor Sum of 11,655

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {11 \, 655} = 23 \, 712$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$11 \, 655 = 3^2 \times 5 \times 7 \times 37$
Hence:
Category:Divisor Sum of Integer
Category:11,655
Divisor Sum of 110

== Example of Divisor Sum of Square-Free Integer ==
:$\map {\sigma_1} {110} = 216$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$110 = 2 \times 5 \times 11$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:110
Divisor Sum of 1184

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1184} = 2394$
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:
:$1184 = 2^5 \times 37$
Hence:
Category:Divisor Sum of Integer
Category:1184
Divisor Sum of 12,285

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {12 \, 285} = 26 \, 880$
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:
:$12 \, 285 = 3^3 \times 5 \times 7 \times 13$
Hence:
Category:Divisor Sum of Integer
Category:12,285
Divisor Sum of 12,496

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {12 \, 496} = 26 \, 784$
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:
:$12 \, 496 = 2^4 \times 11 \times 71$
Hence:
Category:Divisor Sum of Integer
Category:12,496
Divisor Sum of 12,915

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {12 \, 915} = 26 \, 208$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$12 \, 915 = 3^2 \times 5 \times 7 \times 41$
Hence:
Category:Divisor Sum of Integer
Category:12,915
Divisor Sum of 1210

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1210} = 2394$
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:
:$1210 = 2 \times 5 \times 11^2$
Hence:
Category:Divisor Sum of Integer
Category:1210
Divisor Sum of 122,410

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {122 \, 410} = 220 \, 356$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$122 \, 410 = 2 \times 5 \times 12 \, 241$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:122,410
Divisor Sum of 123,228,768

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {123 \, 228 \, 768} = 350 \, 584 \, 416$
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:
:$123 \, 228 \, 768 = 2^5 \times 3 \times 13 \times 293 \times 337$
Hence:
Category:Divisor Sum of Integer
Category:123,228,768
Divisor Sum of 124,015,008

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {124 \, 015 \, 008} = 350 \, 584 \, 416$
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:
:$124 \, 015 \, 008 = 2^5 \times 3 \times 13 \times 99 \, 371$
Hence:
Category:Divisor Sum of Integer
Category:124,015,008
Divisor Sum of 1257

== Example of Divisor Sum of Non-Square Semiprime ==
:$\map {\sigma_1} {1257} = 1680$
== Proof ==
We have that:
:$1257 = 3 \times 419$
and so by definition is a semiprime whose prime factors are distinct.
Hence:
Category:Divisor Sum of Non-Square Semiprime
Category:1257
Divisor Sum of 1260

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1260} = 4368$
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:
:$1260 = 2^2 \times 3^2 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:1260
Divisor Sum of 13,545

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {13 \, 545} = 27 \, 456$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$13 \, 545 = 3^2 \times 5 \times 7 \times 43$
Hence:
Category:Divisor Sum of Integer
Category:13,545
Divisor Sum of 14,175

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {14 \, 175} = 30 \, 008$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$14 \, 175 = 3^4 \times 5^2 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:14,175
Divisor Sum of 14,182,439,040

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {14 \, 182 \, 439 \, 040} = 70 \, 912 \, 195 \, 200$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$14 \, 182 \, 439 \, 040 = 2^7 \times 3^4 \times 5 \times 7 \times 11^2 \times 17 \times 19$
Hence:
Category:Divisor Sum of Integer
Category:14,182,439,040
Divisor Sum of 14,264

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {14 \, 264} = 26 \, 760$
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:
:$14 \, 264 = 2^3 \times 1783$
Hence:
Category:Divisor Sum of Integer
Category:14,264
Divisor Sum of 14,288

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {14 \, 288} = 29 \, 760$
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:
:$14 \, 288 = 2^4 \times 19 \times 47$
Hence:
Category:Divisor Sum of Integer
Category:14,288
Divisor Sum of 14,316

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {14 \, 316} = 33 \, 432$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$14 \, 316 = 2^2 \times 3 \times 1193$
Hence:
Category:Divisor Sum of Integer
Category:14,316
Divisor Sum of 14,536

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {14 \, 536} = 28 \, 800$
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:
:$14 \, 536 = 2^3 \times 23 \times 79$
Hence:
Category:Divisor Sum of Integer
Category:14,536
Divisor Sum of 14,595

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {14 \, 595} = 26 \, 880$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$14 \, 595 = 3 \times 5 \times 7 \times 139$
Hence:
Category:Divisor Sum of Square-Free Integer
Category:Divisor Sum of Integer
Category:14,595
Divisor Sum of 14,805

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {14 \, 805} = 29 \, 952$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$14 \, 805 = 3^2 \times 5 \times 7 \times 47$
Hence:
Category:Divisor Sum of Integer
Category:14,805
Divisor Sum of 140

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {140} = 336$
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:
:$140 = 2^2 \times 5 \times 7$
Hence:
Category:Divisor Sum of Integer
Category:140
Divisor Sum of 1485

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {1485} = 2880$
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:
:$1485 = 3^3 \times 5 \times 11$
Hence:
Category:Divisor Sum of Integer
Category:1485
Divisor Sum of 15,435

== Example of Divisor Sum of Integer ==
:$\map {\sigma_1} {15 \, 435} = 31 \, 200$
where $\sigma_1$ denotes the divisor sum function.
== Proof ==
We have that:
:$15 \, 435 = 3^2 \times 5 \times 7^3$
Hence:
Category:Divisor Sum of Integer
Category:15,435