nyuuzyou/wikis huggingface dataset:
Primitive Recursion on Arithmetically Definable Function is Arithmetically Definable

== Theorem ==
Let $f: \N^k \to \N$ and $g: \N^{k+2} \to \N$ be partial functions.
Let $h: \N^{k + 1} \to \N$ be obtained from $f$ and $g$ by primitive recursion.
Let there exist $\Sigma_1$ WFFs:
:$\map {\phi_f} {y, x_1, \dotsc, x_k}$
:$\map {\phi_g} {y, x_1, \dotsc, x_k, n, z}$
of $k + 1$ and $k + 3$ free variables, respectively, such that:
:$y = \map f {x_1, \dotsc, x_k} \iff \N \models \map {\phi_f} {\sqbrk y, \sqbrk {x_1}, \dotsc, \sqbrk {x_k} }$
:$y = \map g {x_1, \dotsc, x_k, n, z} \iff \N \models \map {\phi_g} {\sqbrk y, \sqbrk {x_1}, \dotsc, \sqbrk {x_k}, \sqbrk n, \sqbrk z}$
where $\sqbrk a$ denotes the unary representation of $a \in \N$.
Then there exists a $\Sigma_1$ WFF of $k + 2$ free variables:
:$\map {\phi_h} {y, x_1, \dotsc, x_k, n}$
such that:
:$y = \map h {x_1, \dotsc, x_k, n} \iff \N \models \map {\phi_h} {\sqbrk y, \sqbrk {x_1}, \dotsc, \sqbrk {x_k}, \sqbrk n}$
== Proof ==
By Gödel's Beta Function is Arithmetically Definable, let:
:$\map {\phi_\beta} {y, a, b, i}$
be a $\Sigma_1$ WFF of $4$ free variables such that:
$y = \map \beta {a, b, i} \iff \N \models \map {\phi_\beta} {\sqbrk y, \sqbrk a, \sqbrk b, \sqbrk i}$
Define $\map {\phi_h} {y, x_1, \dotsc, x_k, n}$ as:
:$\exists a: \exists b: \exists {z_0}: \map {\phi_f} {z_0, x_1, \dotsc, x_k} \land \map {\phi_\beta} {z_0, a, b, 0} \land \map {\phi_\beta} {y, a, b, n} \land \paren {\forall i < n: \exists {z_i}: \exists {z_j}: \map {\phi_\beta} {z_i, a, b, i} \land \map {\phi_\beta} {z_j, a, b, \map s i} \land \map {\phi_g} {z_j, x_1, \dotsc, x_k, i, z_i} }$
To understand this formula, define the finite sequence $\sequence {z_0, \dotsc, z_n}$ defined as:
:$z_i = \map h {x_1, \dotsc, x_k, i}$
By the definition of $h$:
:$z_0 = \map f {x_1, \dotsc, x_k}$
:$z_{i + 1} = \map g {x_1, \dotsc, x_k, i, z_i}$ for all $i < n$
But the formula contains exactly these constraints.
Additionally, as $\phi_\beta$ expresses a function, there is a unique value for the first free variable for which it holds, provided the other variables are held constant.
Then, the formula holds in $\N$ the values for $\map \beta {a, b, i}$ are exactly $\map h {x_1, \dotsc, x_k, i}$.
By Gödel's Beta Function Lemma, there exist $a, b \in \N$ such that:
:$z_i = \map \beta {a, b, i}$
for every $0 \le i \le n$
Therefore, if such $\sequence {z_0, \dotsc, z_n}$ exist, the formula will be true.
Finally, that $\phi_h$ is $\Sigma_1$ follows from:
:Conjunction of Existential Quantifier
:Bounded Universal Quantifier Distributes over Conjunction
Category:Primitive Recursive Functions
Primitive Recursive Function is Total Recursive Function

== Theorem ==
Every primitive recursive function is a total recursive function.
== Proof ==
A primitive recursive function is a total function, which is apparent from its method of definition.
As the processes for generate a primitive recursive function are a subset of those to generate a recursive function, it follows that a primitive recursive function is also a recursive function.
The result follows from the definition of a total recursive function.
Category:Primitive Recursive Functions
Category:Recursive Functions
Primitive Recursive Function is URM Computable

== Theorem ==
Every primitive recursive function‎ is URM computable.
== Proof ==
This follows immediately from:
* The fact that the basic primitive recursive functions are URM computable;
* Functions obtained by substitution from URM computable functions are URM computable;
* Functions obtained by primitive recursion from URM computable functions are URM computable;
* The definition of primitive recursive function‎.
== Note ==
This does not mean that every URM computable function is necessarily primitive recursive.
Category:Primitive Recursive Functions
Primitive Recursive Relation is URM Computable

== Theorem ==
Every primitive recursive relation is URM computable.
== Proof ==
This follows immediately from:
* a relation is primitive recursive if its characteristic function is a primitive recursive
* the fact that every Primitive Recursive Function is URM Computable.
== Note ==
This does not mean that every URM computable relation is necessarily primitive recursive.
Category:Primitive Recursive Functions
Primitive Root is Generator of Reduced Residue System

== Theorem ==
Let $a$ be a primitive root of $n$.
Then:
:$\set {a, a^2, a^3, \ldots, a^{\map \phi n} }$
where $\map \phi n$ is the Euler phi function of $n$, is the reduced residue system of $n$.
Thus the first $\map \phi n$ powers of $a$ "generates" the reduced residue system of $n$.
We say that $a$ is a generator of the reduced residue system of $n$.
== Proof ==
Let $R = \set {a, a^2, a^3, \ldots, a^{\map \phi n} }$.
Each element of $R$ is coprime to $n$ as $a \perp n$.
Suppose $a^r \equiv a^s \pmod n$ where $1 \le r \le s \le \map \phi n$.
Then $a^{r - s} \equiv 1 \pmod n$.
From the definition of primitive root, the multiplicative order of $a$ modulo $n$ is $\map \phi n$.
So from Integer to Power of Multiple of Order $\map \phi n$ divides $r - s$ and so $r = s$.
So no two elements are congruent modulo $n$.
So as $R$ contains $\map \phi n$ integers none of which are congruent modulo $n$ to any of the others, $R$ is a reduced residue system of $n$.
Category:Number Theory
Primitive Semiperfect Number/Examples/104

== Example of Primitive Semiperfect Number ==
$104$ is a primitive semiperfect number:
:$1 + 4 + 8 + 13 + 26 + 52 = 104$
== Proof ==
First it is demonstrated that $104$ is semiperfect.
The aliquot parts of $104$ are enumerated at :
:$1, 2, 4, 8, 13, 26, 52$
$104$ is the sum of a subset of its aliquot parts:
:$1 + 4 + 8 + 13 + 26 + 52 = 104$
Thus $104$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:104
Primitive Semiperfect Number/Examples/1184

== Example of Primitive Semiperfect Number ==
$1184$ is a primitive semiperfect number:
:$1 + 4 + 32 + 37 + 74 + 148 + 296 + 592 = 1184$
== Proof ==
First it is demonstrated that $1184$ is semiperfect.
The aliquot parts of $1184$ are enumerated at :
:$1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592$
$1184$ is the sum of a subset of its aliquot parts:
:$1 + 4 + 32 + 37 + 74 + 148 + 296 + 592$
Thus $1184$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:1184
Primitive Semiperfect Number/Examples/1190

== Example of Primitive Semiperfect Number ==
$1190$ is a primitive semiperfect number:
:$1 + 2 + 5 + 10 + 14 + 17 + 34 + 70 + 85 + 119 + 238 + 595 = 1190$
== Proof ==
First it is demonstrated that $1190$ is semiperfect.
The aliquot parts of $1190$ are enumerated at :
:$1, 2, 5, 7, 10, 14, 17, 34, 35, 70, 85, 119, 170, 238, 595$
$1190$ is the sum of a subset of its aliquot parts:
:$1 + 2 + 5 + 10 + 14 + 17 + 34 + 70 + 85 + 119 + 238 + 595$
Thus $1190$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient except for $70$.
By Semiperfect Number is not Deficient, none of the deficient aliquot parts are themselves semiperfect.
As for $70$ itself, it is seen to be a weird number.
So, by definition, $70$ is not semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:1190
Primitive Semiperfect Number/Examples/20

== Example of Primitive Semiperfect Number ==
$20$ is a primitive semiperfect number:
:$1 + 4 + 5 + 10 = 20$
== Proof ==
First it is demonstrated that $20$ is semiperfect.
The aliquot parts of $20$ are enumerated at :
:$1, 2, 4, 5, 10$
$20$ is the sum of a subset of its aliquot parts:
:$1 + 4 + 5 + 10 = 20$
Thus $20$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:20
Primitive Semiperfect Number/Examples/272

== Example of Primitive Semiperfect Number ==
$272$ is a primitive semiperfect number:
:$1 + 16 + 17 + 34 + 68 + 136 = 272$
== Proof ==
First it is demonstrated that $272$ is semiperfect.
The aliquot parts of $272$ are enumerated at :
:$1, 2, 4, 8, 16, 17, 34, 68, 136$
$272$ is the sum of a subset of its aliquot parts:
:$1 + 16 + 17 + 34 + 68 + 136 = 272$
Thus $272$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:272
Primitive Semiperfect Number/Examples/28

== Example of Primitive Semiperfect Number ==
$28$ is a primitive semiperfect number:
:$1 + 2 + 4 + 7 + 14 = 28$
== Proof ==
We have that $28$ is a perfect number:
:$28 = 1 + 2 + 4 + 7 + 14$
The result follows from Perfect Number is Primitive Semiperfect.
Category:Primitive Semiperfect Numbers/Examples
Category:28
Primitive Semiperfect Number/Examples/304

== Example of Primitive Semiperfect Number ==
$304$ is a primitive semiperfect number:
:$1 + 2 + 16 + 19 + 38 + 76 + 152 = 304$
== Proof ==
First it is demonstrated that $304$ is semiperfect.
The aliquot parts of $304$ are enumerated at :
:$1, 2, 4, 8, 16, 19, 38, 76, 152$
$304$ is the sum of a subset of its aliquot parts:
:$1 + 2 + 16 + 19 + 38 + 76 + 152 = 304$
Thus $304$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:304
Primitive Semiperfect Number/Examples/350

== Example of Primitive Semiperfect Number ==
$350$ is a primitive semiperfect number:
:$1 + 5 + 10 + 14 + 25 + 50 + 70 + 175 = 350$
== Proof ==
First it is demonstrated that $350$ is semiperfect.
The aliquot parts of $350$ are enumerated at :
:$1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175$
$350$ is the sum of a subset of its aliquot parts:
:$1 + 5 + 10 + 14 + 25 + 50 + 70 + 175 = 350$
Thus $350$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient except for $70$.
By Semiperfect Number is not Deficient, none of the deficient aliquot parts are themselves semiperfect.
As for $70$ itself, it is seen to be a weird number.
So, by definition, $70$ is not semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:350
Primitive Semiperfect Number/Examples/368

== Example of Primitive Semiperfect Number ==
$368$ is a primitive semiperfect number:
:$1 + 2 + 4 + 16 + 23 + 46 + 92 + 184 = 368$
== Proof ==
First it is demonstrated that $368$ is semiperfect.
The aliquot parts of $368$ are enumerated at :
:$1, 2, 4, 8, 16, 23, 46, 92, 184$
$368$ is the sum of a subset of its aliquot parts:
:$1 + 2 + 4 + 16 + 23 + 46 + 92 + 184 = 368$
Thus $368$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:368
Primitive Semiperfect Number/Examples/464

== Example of Primitive Semiperfect Number ==
$464$ is a primitive semiperfect number:
:$1 + 4 + 8 + 16 + 29 + 58 + 116 + 232 = 464$
== Proof ==
First it is demonstrated that $464$ is semiperfect.
The aliquot parts of $464$ are enumerated at :
:$1, 2, 4, 8, 16, 29, 58, 116, 232$
$464$ is the sum of a subset of its aliquot parts:
:$1 + 4 + 8 + 16 + 29 + 58 + 116 + 232 = 464$
Thus $464$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:464
Primitive Semiperfect Number/Examples/490

== Example of Primitive Semiperfect Number ==
$490$ is a primitive semiperfect number:
:$2 + 5 + 7 + 14 + 49 + 70 + 98 + 245 = 490$
== Proof ==
First it is demonstrated that $490$ is semiperfect.
The aliquot parts of $490$ are enumerated at :
:$1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245$
$490$ is the sum of a subset of its aliquot parts:
:$2 + 5 + 7 + 14 + 49 + 70 + 98 + 245 = 490$
Thus $490$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient except for $70$.
By Semiperfect Number is not Deficient, none of the deficient aliquot parts are themselves semiperfect.
As for $70$ itself, it is seen to be a weird number.
So, by definition, $70$ is not semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:490
Primitive Semiperfect Number/Examples/496

== Example of Primitive Semiperfect Number ==
$496$ is a primitive semiperfect number:
:$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496$
== Proof ==
We have that $496$ is a perfect number:
:$496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$
The result follows from Perfect Number is Primitive Semiperfect.
Category:Primitive Semiperfect Numbers/Examples
Category:496
Primitive Semiperfect Number/Examples/550

== Example of Primitive Semiperfect Number ==
$550$ is a primitive semiperfect number:
:$2 + 11 + 22 + 25 + 50 + 55 + 110 + 275 = 550$
== Proof ==
First it is demonstrated that $550$ is semiperfect.
The aliquot parts of $550$ are enumerated at :
:$1, 2, 5, 10, 11, 22, 25, 50, 55, 110, 275$
$550$ is the sum of a subset of its aliquot parts:
:$2 + 11 + 22 + 25 + 50 + 55 + 110 + 275 = 550$
Thus $550$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:550
Primitive Semiperfect Number/Examples/572

== Example of Primitive Semiperfect Number ==
$572$ is a primitive semiperfect number:
:$1 + 11 + 13 + 22 + 44 + 52 + 143 + 286 = 572$
== Proof ==
First it is demonstrated that $572$ is semiperfect.
The aliquot parts of $572$ are enumerated at :
:$1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286$
$572$ is the sum of a subset of its aliquot parts:
:$1 + 11 + 13 + 22 + 44 + 52 + 143 + 286 = 572$
Thus $572$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:572
Primitive Semiperfect Number/Examples/6

== Example of Primitive Semiperfect Number ==
$6$ is a primitive semiperfect number:
:$1 + 2 + 3 = 6$
== Proof ==
We have that $6$ is a perfect number:
:$6 = 1 + 2 + 3$
The result follows from Perfect Number is Primitive Semiperfect.
Category:Primitive Semiperfect Numbers/Examples
Category:6
Primitive Semiperfect Number/Examples/650

== Example of Primitive Semiperfect Number ==
$650$ is a primitive semiperfect number:
:$1 + 5 + 10 + 13 + 25 + 26 + 50 + 65 + 130 + 325 = 650$
== Proof ==
First it is demonstrated that $650$ is semiperfect.
The aliquot parts of $650$ are enumerated at :
:$1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325$
$650$ is the sum of a subset of its aliquot parts:
:$1 + 5 + 10 + 13 + 25 + 26 + 50 + 65 + 130 + 325 = 650$
Thus $650$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:650
Primitive Semiperfect Number/Examples/748

== Example of Primitive Semiperfect Number ==
$748$ is a primitive semiperfect number:
:$2 + 17 + 22 + 34 + 44 + 68 + 187 + 374 = 748$
== Proof ==
First it is demonstrated that $748$ is semiperfect.
The aliquot parts of $748$ are enumerated at :
:$1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374$
$748$ is the sum of a subset of its aliquot parts:
:$2 + 17 + 22 + 34 + 44 + 68 + 187 + 374 = 748$
Thus $748$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:748
Primitive Semiperfect Number/Examples/770

== Example of Primitive Semiperfect Number ==
$770$ is a primitive semiperfect number:
:$1 + 5 + 7 + 11 + 14 + 35 + 55 + 70 + 77 + 110 + 385 = 770$
== Proof ==
First it is demonstrated that $770$ is semiperfect.
The aliquot parts of $770$ are enumerated at :
:$1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385$
$770$ is the sum of a subset of its aliquot parts:
:$1 + 5 + 7 + 11 + 14 + 35 + 55 + 70 + 77 + 110 + 385$
Thus $770$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient except for $70$.
By Semiperfect Number is not Deficient, none of the deficient aliquot parts are themselves semiperfect.
As for $70$ itself, it is seen to be a weird number.
So, by definition, $70$ is not semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:770
Primitive Semiperfect Number/Examples/88

== Example of Primitive Semiperfect Number ==
$88$ is a primitive semiperfect number:
:$1 + 2 + 8 + 11 + 22 + 44 = 88$
== Proof ==
First it is demonstrated that $88$ is semiperfect.
The aliquot parts of $88$ are enumerated at :
:$1, 2, 4, 8, 11, 22, 44$
$88$ is the sum of a subset of its aliquot parts:
:$1 + 2 + 8 + 11 + 22 + 44 = 88$
Thus $88$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:88
Primitive Semiperfect Number/Examples/910

== Example of Primitive Semiperfect Number ==
$910$ is a primitive semiperfect number:
:$1 + 2 + 5 + 7 + 10 + 13 + 26 + 35 + 65 + 70 + 91 + 130 + 455 = 910$
== Proof ==
First it is demonstrated that $910$ is semiperfect.
The aliquot parts of $910$ are enumerated at :
:$1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455$
$910$ is the sum of a subset of its aliquot parts:
:$1 + 2 + 5 + 7 + 10 + 13 + 26 + 35 + 65 + 70 + 91 + 130 + 455$
Thus $910$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient except for $70$.
By Semiperfect Number is not Deficient, none of the deficient aliquot parts are themselves semiperfect.
As for $70$ itself, it is seen to be a weird number.
So, by definition, $70$ is not semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:910
Primitive Semiperfect Number/Examples/945

== Example of Primitive Semiperfect Number ==
$945$ is a primitive semiperfect number:
:$1 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 945$
== Proof ==
First it is demonstrated that $945$ is semiperfect.
The aliquot parts of $945$ are enumerated at :
:$1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315$
$945$ is the sum of a subset of its aliquot parts:
:$1 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 63 + 105 + 135 + 189 + 315$
Thus $945$ is semiperfect by definition.
By inspecting the divisor sums of each of those aliquot parts, they are seen to be deficient.
By Semiperfect Number is not Deficient, none of these are themselves semiperfect.
Hence the result, by definition of primitive semiperfect number.
Category:Primitive Semiperfect Numbers/Examples
Category:945
Primitive Semiperfect Numbers which are not Primitive Abundant

== Theorem ==
The sequence of primitive semiperfect numbers which are not also primitive abundant starts:
:$6, 28, 350, 490, 496, 770, 910, 1190, \ldots$
These are semiperfect numbers which are either:
: perfect
or:
: whose only abundant aliquot parts are weird.
== Proof ==
A primitive semiperfect number is a semiperfect number which has no aliquot parts which are themselves semiperfect.
Thus by definition a primitive semiperfect number is either perfect or abundant.
A primitive abundant number is an abundant number whose aliquot parts are all deficient.
Thus the perfect numbers:
:$6, 28, 496, \ldots$
are not primitive abundant.
However, by Divisor of Perfect Number is Deficient, the perfect numbers are all by definition primitive semiperfect.
Hence the presence of the perfect numbers in this sequence.
Next, consider the sequence of weird numbers:
:$70, 836, \ldots$
These are numbers which are abundant but not semiperfect.
Thus an abundant number whose aliquot parts are all deficient except for one or more weird numbers is a primitive semiperfect number which is not primitive abundant.
However, such a number is not primitive abundant because it has an aliquot part which is abundant, though weird.
So:
:while $70$ is primitive abundant it is not primitive semiperfect number
:while $350$, $490$, $710$, $910$, $1190$, and so on, are primitive semiperfect, they are not primitive abundant.
Hence the result.
Category:Primitive Semiperfect Numbers
Category:Primitive Abundant Numbers
Primitive of Arccosecant of x over a

== Theorem ==
:$\ds \int \arccsc \frac x a \rd x = \begin {cases}
x \arccsc \dfrac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arccsc \dfrac x a < \dfrac \pi 2 \\
x \arccsc \dfrac x a - a \map \ln {x + \sqrt {x^2 - a^2} } + C & : -\dfrac \pi 2 < \arccsc \dfrac x a < 0 \\
\end {cases}$
== Proof ==
With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
and let:
First let $\arccsc \dfrac x a$ be in the interval $\openint 0 {\dfrac \pi 2}$.
Then:
Similarly, let $\arccsc \dfrac x a$ be in the interval $\openint {-\dfrac \pi 2} 0$.
Then:
== Also see ==
* Primitive of $\arcsin \dfrac x a$
* Primitive of $\arccos \dfrac x a$
* Primitive of $\arctan \dfrac x a$
* Primitive of $\arccot \dfrac x a$
* Primitive of $\arcsec \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.498$
* : Appendix $2$: Table of derivatives and integrals of common functions: Inverse trigonometric functions
Category:Primitives involving Inverse Cosecant Function
Category:Arccosecant Function
Primitive of Arccosecant of x over a over x

== Theorem ==
== Proof ==
== Also see ==
* Primitive of $\dfrac 1 x \arcsin \dfrac x a$
* Primitive of $\dfrac 1 x \arccos \dfrac x a$
* Primitive of $\dfrac 1 x \arctan \dfrac x a$
* Primitive of $\dfrac 1 x \arccot \dfrac x a$
* Primitive of $\dfrac 1 x \arcsec \dfrac x a$
== Sources ==
* : $\S 14$: Integrals involving Inverse Trigonometric Functions: $14.501$
Category:Primitives involving Inverse Cosecant Function