nyuuzyou/wikis huggingface dataset:
10,112,359,550,561,797,752,808,988,764,044,943,820,224,719

== Number ==
$10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$ is:
:$3^4 \times 11^2 \times 23 \times 101 \times 4093 \times 8779 \times 21 \, 649 \times 513 \, 239 \times 1 \, 052 \, 788 \, 969 \times 1 \, 056 \, 689 \, 261$
:The smallest positive integer $N$ such that if you move the last digit to the front, the result is the positive integer $9 N$:
::$10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719 \times 9 = 91 \, 011 \, 235 \, 955 \, 056 \, 179 \, 775 \, 280 \, 898 \, 876 \, 404 \, 494 \, 382 \, 022 \, 471$
== Also see ==
* Integer which is Multiplied by 9 when moving Last Digit to First
== Sources ==
* : $10,112,359,550,561,797,752,808,988,764,044,943,820,224,719$
Category:Specific Numbers
Category:10,112,359,550,561,797,752,808,988,764,044,943,820,224,719
10,206

== Number ==
$10 \, 206$ (ten thousand, two hundred and six) is:
:$2 \times 3^6 \times 7$
:The $27$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $\ldots$, $1470$, $1800$, $2176$, $2601$, $3078$, $3610$, $4200$, $4851$, $5566$, $6348$, $7200$, $8125$, $9126$:
::$10 \, 206 = \ds \sum_{k \mathop = 1}^{27} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {27^2 \paren {27 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:10,206
10,213,223

== Number ==
$10 \, 213 \, 223$ (ten million, two hundred and thirteen thousand, two hundred and twenty-three) is:
:$41 \times 249103$
:A fixed point in the ascending pea pattern:
::$10213223$ is one $0$, two $1$s, three $2$s, two $3$s
== Also see ==
* Ascending Pea Pattern on $10 \, 213 \, 223$
== Sources ==
* : $10,213,223$
Category:Specific Numbers
Category:10,213,223
10,585

== Number ==
$10 \, 585$ (ten thousand, five hundred and eighty-five) is:
:$5 \times 29 \times 73$
:The $8$th Carmichael number after $561$, $1105$, $1729$, $2465$, $2821$, $6601$, $8911$:
::$\forall a \in \Z: a \perp 10 \, 585: a^{10 \, 584} \equiv 1 \pmod {10 \, 585}$
== Also see ==
* 
Category:Specific Numbers
Category:10,585
10,662,526,601

== Number ==
$10 \, 662 \, 526 \, 601$ is:
:$31^3 \times 71^3$
:The $2201$st cube number:
::$10 \, 662 \, 526 \, 601 = 2201 \times 2201 \times 2201$
:The $8$th palindromic cube after $1$, $8$, $343$, $1331$, $1 \, 030 \, 301$, $1 \, 367 \, 631$, $1 \, 003 \, 003 \, 001$:
::$10 \, 662 \, 526 \, 601 = 2201^3$
:The only known palindromic cube with a root that is not itself palindromic
== Also see ==
* Palindromic Cube with Non-Palindromic Root
* 
* 
== Sources ==
* : $10,662,526,601$
* : $10,662,526,601$
Category:Specific Numbers
Category:10,662,526,601
10,744

== Number ==
$10 \, 744$ (ten thousand, seven hundred and forty-four) is:
:$2^3 \times 17 \times 79$
:The smaller of the $6$th amicable pair, with $10 \, 856$:
::$\map {\sigma_1} {10 \, 744} = \map {\sigma_1} {10 \, 856} = 21 \, 600 = 10 \, 744 + 10 \, 856$
== Also see ==
* 
Category:Specific Numbers
Category:10,744
10,856

== Number ==
$10 \, 856$ (ten thousand, eight hundred and fifty-six) is:
:$2^3 \times 23 \times 59$
:The smaller of the $6$th amicable pair, with $10 \, 744$:
::$\map {\sigma_1} {10 \, 744} = \map {\sigma_1} {10 \, 856} = 21 \, 600 = 10 \, 744 + 10 \, 856$
== Also see ==
* 
Category:Specific Numbers
Category:10,856
10,989

== Number ==
$10,989$ (ten thousand, nine hundred and eighty-nine) is:
:$3^3 \times 11 \times 37$
:The $2$nd number which is reversed when multiplied by $9$:
::$10 \, 989 \times 9 = 98 \, 901$
:The $1$st of only $2$ numbers with $5$ digits or which has a multiple which is its reversal:
::$10 \, 989 \times 9 = 98 \, 901$
== Also see ==
* 
* 
== Sources ==
* : $10,989$
* : $10,989$
Category:Specific Numbers
Category:10,989
10/Historical Note

== Historical Note on $10$ (Ten) ==
 made the obvious statement that $10$ is the usual number base because we have $10$ fingers. This view was echoed by the Roman poet .
It is noted that not all cultures use $10$ -- some use $5$ (based on the number of fingers on a single hand) and some use $20$ (based on the total number of fingers and toes).
The number $10$ was considered holy by the Pythagoreans, who set great score to the fact that $10 = 1 + 2 + 3 + 4$.
They knew about $8$ celestial bodies:
:Sun, Moon, Mercury, Venus, Earth, Mars, Jupiter, Saturn
and also believed in the .
In order to satisfy their belief that $10$ was such a special number, they hypothesized the existence of a which orbited the in such a position as to be hidden from us by the Sun.
The number $10$ is expressed in Roman numerals as $\mathrm X$.
It has been suggested that this originates from:
:joining up two $\mathrm V$s, each representing $2$ hands held up showing the full $5$ fingers
or:
:an abbreviation for $10$ tally marks, with a line struck through them to indicate that $10$ had been reached.
== Also see ==
* Definition:Tetractys
== Sources ==
* : $10$
* : $50$
* : $10$
* : $50$
* : Chapter $2$: The Logic of Shape: Pythagoras
Category:Historical Notes
100

== Number ==
$100$ (one hundred or a hundred) is:
:$2^2 \times 5^2$
:The sum of the first $4$ cubes:
::$100 = 1^3 + 2^3 + 3^3 + 4^3$
:The $2$nd power of $10$ after $(1)$, $10$:
::$100 = 10^2$
:The $4$th after $2$, $5$ and $10$ of $4$ numbers whose letters, when spelt in French, are in alphabetical order:
::cent
:The $8$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$:
::$100 = \dfrac {8 \paren {3 \times 8 + 1} } 2$
:The $8$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 100$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $9$ different ways
:The $10$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$:
::$100 = 10 \times 10$
:The $14$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$
:The $16$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$, $51$, $57$, $70$, $77$, $92$:
::$100 = \dfrac {8 \paren {3 \times 8 + 1} } 2$
:The $20$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $28$, $31$, $32$, $44$, $49$, $68$, $70$, $79$, $82$, $86$, $91$, $94$, $97$:
::$100 \to 1^2 + 0^2 + 0^2 = 1$
:The $23$rd semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$, $66$, $72$, $78$, $80$, $84$, $88$, $90$, $96$:
::$100 = 5 + 20 + 25 + 50$
:The $49$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $61$, $65$, $66$, $67$, $72$, $77$, $80$, $81$, $84$, $89$, $94$, $95$, $96$ which cannot be expressed as the sum of distinct pentagonal numbers.
== Also see ==
* 100 using Digits from 1 to 9
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Linguistic Note ==
== Sources ==
* : $100$
* : $100$
Category:Specific Numbers
Category:100
100,000

== Number ==
$100 \, 000$ (one hundred thousand) is:
:$2^5 \times 5^5$
:The $10$th fifth power after $1$, $32$, $243$, $1024$, $3125$, $7776$, $16 \, 807$, $32 \, 768$, $59 \, 049$:
::$100 \, 000 = 10 \times 10 \times 10 \times 10 \times 10$
:The $5$th power of $10$ after $(1)$, $10$, $100$, $1000$, $10 \, 000$:
::$100 \, 000 = 10^5$
== Also see ==
* 
* 
Category:Specific Numbers
Category:100,000
100,895,598,169

== Number ==
$100\,895\,598\,169$ is:
:$112\,303 \times 898\,423$
:One of the divisors of a huge integer that challenged to analyse.
== Also see ==
=== Multiply Perfect Number of Order 6: Historical Note ===
== Sources ==
* : $100,895,598,169$
* : Chapter $\text {A}.13$: Fermat ()
* : $100,895,598,169$
Category:Specific Numbers
Category:100,895,598,169
100/Historical Note

== Historical Note on $100$ (One Hundred) ==
The number $100$ (one hundred) was at one time sometimes referred to in England as a short hundred.
This was to distinguish it from the term long hundred for the number $120$.
Both terms are now obsolete in England, although the term great hundred for $120$ is still used in Germany and Scandinavia.
The boiling point of water is defined as being $100$ degrees Celsius.
The number $100$ is expressed in Roman numerals as $\mathrm C$.
This originates from the first letter of the Latin word centum, meaning $100$.
The archetypal "big number", to small children, is $100$. This most likely stems from the fact that their first introduction to numbers is the exercise to count to $100$. As this is where the count stops, $100$ is the biggest number they know.
Once introduced to the concept of one hundred and one, however, their level of arithmetical sophistication is soon seen to increase.
== Sources ==
* : $12$
* : $50$
* : $100$
* : $12$
* : $50$
* : $100$
Category:Historical Notes
1000

== Number ==
$1000$ (one thousand or a thousand) is:
:$2^3 \times 5^3$
:The $3$rd power of $10$ after $(1)$, $10$, $100$:
::$1000 = 10^3$
:The $10$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$:
::$1000 = 10 \times 10 \times 10$
:$1000 = 10^3$ in any base whatsoever.
== Also see ==
* Number of Binary Digits in Power of 10/Example/1000
* 
* 
* 
== Historical Note ==
== Sources ==
* : $1000$
* : $1000$
Category:Specific Numbers
Category:1000
1001

== Number ==
$1001$ (one thousand and one) is:
:$7 \times 11 \times 13$
:The $4$th pentagonal number after $1$, $5$, $22$ which is also palindromic:
::$1001 = \ds \sum_{k \mathop = 1}^{26} \paren {3 k - 2} = \dfrac {26 \paren {3 \times 26 - 1} } 2$
:The $7$th positive integer after $1$, $2$, $7$, $11$, $101$, $111$ whose cube is palindromic:
::$1001^3 = 1 \, 003 \, 003 \, 001$
:The $11$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$, $495$, $715$:
::$1001 = \ds \sum_{k \mathop = 1}^{11} \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {11 \paren {11 + 1} \paren {11 + 2} \paren {11 + 3} } {24}$
:The $12$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $111$, $121$, $202$, $212$ whose square is also palindromic integer
::$1001^2 = 1 \, 002 \, 001$
:The $26$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $477$, $532$, $590$, $651$, $715$, $782$, $852$, $925$:
::$1001 = \ds \sum_{k \mathop = 1}^{26} \paren {3 k - 2} = \dfrac {26 \paren {3 \times 26 - 1} } 2$
:The $51$st generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$, $876$, $925$, $950$:
::$1001 = \ds \sum_{k \mathop = 1}^{26} \paren {3 k - 2} = \dfrac {26 \paren {3 \times 26 - 1} } 2$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $1001$
* : $1001$
Category:Specific Numbers
Category:1001
1002

== Number ==
$1002$ (one thousand and two) is:
:$2 \times 3 \times 167$
:The number of integer partitions for $22$:
::$\map p {22} = 1002$
:The $211$th nontotient after $992$, $994$, $998$:
::$\nexists m \in \Z_{>0}: \map \phi m = 1002$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:1002
1008

== Number ==
$1008$ (one thousand and eight) is:
:$2^4 \times 3^2 \times 7$
:The $4$th positive integer after $504$, $756$, $806$ which can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other:
::$1008 = 12 \times 84 = 48 \times 21$
:The $44$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $336$, $360$, $420$, $480$, $504$, $600$, $630$, $660$, $720$, $840$, $960$:
::$\map {\sigma_1} {1008} = 3224$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1008
1009

== Number ==
$1009$ (one thousand and nine) is:
:The $169$th prime number
:The $1$st prime number which can be expressed in the form $x^2 + n y^2$ for all values of $n$ from $1$ to $10$:
:The $34$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $733$, $739$, $743$, $751$, $761$, $769$, $907$, $937$, $941$, $953$, $967$, $971$, $983$, $991$
== Also see ==
* 
* 
* 
== Sources ==
* : $1009$
Category:Specific Numbers
Category:1009
100 in Golden Mean Number System is Equivalent to 011

== Theorem ==
Consider the golden mean number system.
Let $p$ and $q$ be arbitrary strings in $\set {0, 1}$.
Let $x \in \R_{\ge 0}$ have a representation which includes the string $100$, say:
:$x = p100q$
Then $x \in \R_{\ge 0}$ also has the representation:
:$x = p011q$
Similarly, let $x \in \R_{\ge 0}$ have a representation which includes the string $011$, say:
:$x = p011q$
Then $x \in \R_{\ge 0}$ also has the representation:
:$x = p100q$
That is, any instance of $100$ appearing in a representation of a non-negative real number $x$ is equivalent to $011$, and vice versa.
Note that the instance of $100$ or $011$ may also include a radix point; the instance of $011$ or $100$ to which it is equivalent will include the radix point in the same location.
== Proof ==
Let $100$ appear anywhere within $x$.
Then:
:$x = \phi^r + \ds \sum_{c \mathop \in C} \phi^c$
where:
:$C \subset \Z$
:$r \in \Z$
:$r \notin C, r - 1 \notin C, r - 2 \notin C$
That is, the instance of $100$ corresponds to the indices $r, r - 1, r - 2$.
From Power of Golden Mean as Sum of Smaller Powers:
:$\phi^r = \phi^{r - 1} + \phi^{r - 2}$
and so:
:$x = \phi^{r - 1} + \phi^{r - 2} + \ds \sum_{c \mathop \in C} \phi^c$
That is, the indices instance of $100$ corresponds to the indices $r, r - 1, r - 2$ now correspond to the string $011$.
== Sources ==
* 
Category:Golden Mean Number System
100 using Digits from 1 to 9

== Classic Problem ==
Using just the digits $1$ to $9$ in order (ascending and descending), using the usual operations ($+$, $-$, $\times$, $\div$) and brackets, how can you make $100$?
== Solution ==
There are many answers:
:$1 + 2 + 3 + 4 + 5 + 6 + 7 + \paren {8 \times 9} = 100$
:$123 - 45 - 67 + 89 = 100$
Using the digits in reverse order:
:$98 - 76 + 54 + 3 + 21 = 100$
... and of course the initial answer:
:$\paren {9 \times 8} + 7 + 6 + 5 + 4 + 3 + 2 + 1$
== Historical Note ==
== Sources ==
* : $100$
* : $100$
Category:Classic Problems
Category:100
100 using Four 9s/Solution 2

== Puzzle ==
== Solution ==
:$100 = \paren {9 + .\dot 9} \paren {9 + .\dot 9}$
where $.\dot 9$ is interpreted to mean point $9$ recurring:
:$.\dot 9 := .9999 \ldots = 1$
Category:100 using Four 9s
101

== Number ==
$101$ (one hundred and one) is:
:The $26$th prime number
:The $1$st near-repdigit prime
:The $3$rd prime number after $5$, $19$ of the form $\ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$:
::$101 = 5! - 4! + 3! - 2! + 1!$
:The $4$th unique period prime after $3$, $11$, $37$: its period is $4$:
::$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$
:The smallest positive integer the decimal expansion of whose reciprocal has a period of $4$:
::$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$
:The upper end of the $5$th record-breaking gap between twin primes:
::$101 - 73 = 28$
:The $5$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$:
::$101 = 10^2 + 1$
:The $5$th positive integer after $1$, $2$, $7$, $11$ whose cube is palindromic:
::$101^3 = 1 \, 030 \, 301$
:The $6$th palindromic prime after $2$, $3$, $5$, $7$, $11$
:The $7$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$ whose square is also palindromic integer
::$101^2 = 10 \, 201$
:The $7$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 6^2 + 29 = 101$ ()
:The smaller of the $9$th pair of twin primes, with $103$
:The $12$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$, $85$, $89$ such that the Fibonacci number $F_n$ ends in $n$
:The number of integer partitions for $13$:
::$\map p {13} = 101$
:The $36$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $89$, $97$, $101$, $\ldots$
:The $50$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $61$, $65$, $66$, $67$, $72$, $77$, $80$, $81$, $84$, $89$, $94$, $95$, $96$, $100$ which cannot be expressed as the sum of distinct pentagonal numbers
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $101$
* : $101$
Category:Specific Numbers
Category:101
101/Historical Note

== Historical Note on $101$ (One Hundred and One) ==
The number $101$ has several cultural significances.
* The rhetorical device meaning: a large number ($100$) and then some (and $1$), often in the context of book titles:
::, and , and so on.
* Room $101$: In by , room $101$ is the name of the torture chamber where bad things happen. It has entered contemporary consciousness for a rhetorical and metaphorical space into which things are to be consigned which are particularly hated.
* As a metaphor for an entry-level course of study. It is usual, in Western universities, for the first course in the first year of study of a degree, to be provided with the serial number xx$101$, where xx identifies the nature of the study course. Hence it is used rhetorically, often for the purposes of ridicule, to mean basic information which you really ought to know by now.
::Oh come on, forgetting to set the timer? That's cooking $101$!
Category:Historical Notes
1013

== Number ==
$1013$ (one thousand and thirteen) is:
:The $170$th prime number
:The $38$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $653$, $659$, $683$, $719$, $743$, $761$, $809$, $911$, $953$:
::$2 \times 1013 + 1 = 2027$, which is prime.
== Also see ==
* 
* 
Category:Specific Numbers
Category:1013
1015

== Number ==
$1015$ (one thousand, one hundred and fifteen) is:
:$5 \times 7 \times 29$
:The $14$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $91$, $140$, $204$, $285$, $385$, $506$, $650$, $819$:
::$1015 = \ds \sum_{k \mathop = 1}^{14} k^2 = \dfrac {14 \paren {14 + 1} \paren {2 \times 14 + 1} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:1015
1019

== Number ==
$1019$ (one thousand and nineteen) is:
:The $171$st prime number
:The $8$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$
:The $20$th obstinate number after $1$, $127$, $149$, $251$, $331$, $337$, $373$, $509$, $599$, $701$, $757$, $757$, $809$, $877$, $905$, $907$, $959$, $977$, $997$
:The $26$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$, $479$, $503$, $563$, $587$, $719$, $839$, $863$, $887$, $983$:
::$1019 = 2 \times 509 + 1$
:The smaller of the $36$th pair of twin primes:
::$1021 - 1019 = 2$
:The $39$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $653$, $659$, $683$, $719$, $743$, $761$, $809$, $911$, $953$, $1013$:
::$2 \times 1019 + 1 = 2039$, which is prime.
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:1019
101 is Smallest Number whose Period of Reciprocal is 4

== Theorem ==
$101$ is the first positive integer the decimal expansion of whose reciprocal has a period of $4$:
:$\dfrac 1 {101} = 0 \cdotp \dot 009 \dot 9$
== Proof ==
Let the positive integer reciprocal be $\dfrac1 k$ for some $k \in \Z_{\ge 0}$.
For it to have a period of recurrence of $4$ in base $10$, it must be able to be expressed as $\dfrac a {10^4-1}$ for some $a \in \Z_{\ge 0}$. 
Since this rational number must have a period of recurrence minimum of $4$, it cannot also have a period of $2$ or $1$. It will be sufficient for this proof to only consider periods of $2$. This rational number must not be able to be expressed as $\dfrac b {10^2-1}$ for any $b \in \Z_{\ge 0}$.
----
$\dfrac1 k \neq \dfrac b {10^2-1}$
$\leadsto \dfrac a {99 \cdot 101} \neq \dfrac b {99}$
$\leadsto a \neq 101 b$
Since $101$ is prime, $a$ is not a multiple of $101$ but $ak$ is, then $k$ must be a multiple of $101$. $k = 101$ is the first instance which satisfies these conditions and produces a reciprocal with a period of recurrence of 4.
Category:101
Category:Examples of Reciprocals
102

== Number ==
$102$ (one hundred and two) is:
:$2 \times 3 \times 17$
:The $3$rd positive integer after $1$, $7$ the sum of whose divisors is a cube:
::$\map {\sigma_1} {102} = 216 = 6^3$
:It is also the $2$nd of those positive integer after $1$ whose divisor count is also a cube:
::$\map {\sigma_0} {102} = 8 = 2^3$
:The $6$th sphenic number after $30$, $42$, $66$, $70$, $78$:
::$102 = 2 \times 3 \times 17$
:The smallest positive integer whose $7$th power can be expressed as the sum of $8$ other $7$th powers:
::$102^7 = 12^7 + 35^7 + 53^7 + 58^7 + 64^7 + 83^7 + 85^7 + 90^7$
:The $51$st positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $61$, $65$, $66$, $67$, $72$, $77$, $80$, $81$, $84$, $89$, $94$, $95$, $96$, $100$, $101$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $52$nd (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$, $84$, $90$, $91$, $95$, $96$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* Smallest Seventh Power which is Sum of 8 other Seventh Powers
* 
* 
* 
* 
== Sources ==
* : $102$
* : $102$
Category:Specific Numbers
Category:102
1021

== Number ==
$1021$ (one thousand and twenty-one) is:
:The $172$nd prime number
:The $9$th prime $p$ such that $p \# + 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$2$, $3$, $5$, $7$, $11$, $31$, $379$, $1019$, $1021$
:The smaller of the $17$th pair of primes whose prime gap is $10$:
::$1031 - 1021 = 10$
:The $35$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $733$, $739$, $743$, $751$, $761$, $769$, $907$, $937$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$
:The larger of the $36$th pair of twin primes:
::$1021 - 1019 = 2$
== Also see ==
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Category:Specific Numbers
Category:1021
1024

== Number ==
$1024$ (one thousand and twenty-four) is:
:$2^{10}$
:The $1$st number with at least $10$ prime factors:
::$1024 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
:The $4$th fifth power after $1$, $32$, $243$:
::$1024 = 4 \times 4 \times 4 \times 4 \times 4$
:The $5$th power of $4$ after $(1)$, $4$, $16$, $64$, $256$:
::$1024 = 4^5$
:The $10$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$:
::$1024 = 2^{10}$
:The $11$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$:
::$\map {\sigma_1} {1024} = 2047 = 2 \times 1024 - 1$
:The $13$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$, $400$, $576$, $784$, $900$, $961$ to be the divisor sum value of some (strictly) positive integer:
::$1024 = \map {\sigma_1} {651} = \map {\sigma_1} {889}$
:The $28$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $400$, $526$, $529$, $625$, $676$, $706$, $730$, $771$, $784$, $841$, $1024$, $\ldots$
:The $32$nd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $625$, $676$, $729$, $784$, $841$, $900$, $961$:
::$1024 = 32 \times 32$
== Also see ==
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== Historical Note ==
== Sources ==
* : $1024$
* : $1024$
Category:Specific Numbers
Category:1024
1024/Historical Note

== Historical Note on $1024$ (One Thousand and Twenty-Four) ==
While the metric scaling prefix $\mathrm k$ is generally used to denote a scale factor of $1000$, in the field of computer science it is used to mean a scale factor of $1024$.
In order to distinguish the two in case of confusion, the $1024$ multiplier can be denoted by an uppercase $\mathrm K$.
Thus $1 \, \mathrm K$ of memory conventionally means $1024$ bytes.
It is of course purely coincidental that $2^{10}$ is very close in value to $10^3$.
== Sources ==
* : $1024$
* : $1024$
Category:Historical Notes