nyuuzyou/wikis huggingface dataset:
2977

== Number ==
$2977$ (two thousand, nine hundred and seventy-seven) is:
:$13 \times 229$
:The $32$nd centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $1387$, $1519$, $1657$, $1801$, $1951$, $2107$, $2269$, $2437$, $2611$, $2791$:
::$2977 = \ds 1 + \sum_{k \mathop = 1}^{32 - 1} 6 k = 32^3 - 31^3$
== Also see ==
* 
Category:Specific Numbers
Category:2977
298

== Number ==
$298$ (two hundred and ninety-eight) is:
:$2 \times 149$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $17$ different ways
:The $30$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $244$, $260$, $266$, $268$, $274$, $290$, $292$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 298$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $48$th nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 298$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:298
2986

== Number ==
$2986$ (two thousand, nine hundred and eighty-six) is:
:$2 \times 1493$
:The $18$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $91$, $130$, $214$, $226$, $370$, $526$, $706$, $730$, $771$, $1255$, $1351$, $1414$, $1906$, $2986$, $\ldots$
:The $45$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $1521$, $1681$, $1849$, $1906$, $1936$, $2116$, $2209$, $2304$, $2500$, $2809$, $2986$, $\ldots$
== Also see ==
* 
* 
Category:Specific Numbers
Category:2986
2 to the n is Greater than n Cubed when n is 10 and above

== Theorem ==
:$\forall n \in \Z, n \ge 10: 2^n > n^3$
== Proof ==
Proof by induction:
For all $n \in \Z$ such that $n \ge 10$, let $\map P n$ be the proposition:
:$2^n > n^3$
We note that:
:$2^9 = 512 < 729 = 9^3$
so when $n < 10$ the proposition does not hold.
=== Basis for the Induction ===
$\map P {10}$ is the case:
:$2^{10} = 1024 > 1000 = 10^3$
so $\map P {10}$ is seen to hold.
This is our basis for the induction.
=== Induction Hypothesis ===
Now we need to show that, if $\map P k$ is true, where $k \ge 10$, then it logically follows that $\map P {k + 1}$ is true.
So this is our induction hypothesis:
:$2^k > k^3$
We need to show that:
:$2^{k + 1} > \paren {k + 1}^3$
=== Induction Step ===
This is our induction step:
We note that when $k \ge 10$:
Thus:
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
== Sources ==
* : $\S 1.2.1$: Mathematical Induction: Exercise $10$
Category:Number Theory
Category:2
Category:10
3,010,349

== Number ==
$3 \, 010 \, 349$ (three million, ten thousand, three hundred and forty-nine) is:
:The $217 \, 506$th prime number
:The $12$th Lucas prime after $2$, $3$, $7$, $11$, $29$, $47$, $199$, $521$, $2207$, $3571$, $9349$.
== Also see ==
* 
* 
Category:Specific Numbers
Category:3,010,349
3,285,697,041

== Number ==
$3 \, 285 \, 697 \, 041$ is:
:$3^6 \times 11^2 \times 193^2$
:The $57 \, 321$st square number:
::$3 \, 285 \, 697 \, 041 = 57 \, 321 \times 57 \, 321$
:The square of the smaller of the $2$nd pandigital pair, with $60 \, 984$, of integers each have squares which are themselves pandigital:
::$57 \, 321^2 = 3 \, 285 \, 697 \, 041$, $60 \, 984^2 = 3 \, 719 \, 048 \, 256$
== Also see ==
* Pandigital Pairs whose Squares are Pandigital
== Sources ==
* : $57,321$
Category:Specific Numbers
Category:3,285,697,041
3,424,506

== Number ==
$3 \, 424 \, 506$ (three million, four hundred and twenty-four thousand, five hundred and six) is:
:$2 \times 3 \times 101 \times 5651$
:The number of pairs of twin primes less than $10^9$
== Also see ==
* 
== Sources ==
* : $3,424,506$
Category:Specific Numbers
Category:3,424,506
3,430,751,869

== Number ==
$3\,430\,751\,869$ is:
:A prime number
:The $1$st of the smallest $17$ primes in arithmetic sequence:
::$3\,430\,751\,869 = 3\,430\,751\,869 + 0 \times 87\,297\,210$
== Also see ==
* 
* Smallest 17 Primes in Arithmetic Sequence
== Sources ==
* : $3,430,751,869$
Category:Specific Numbers
Category:3,430,751,869
3,544,453

== Number ==
$3 \, 544 \, 453$ (three million, five hundred and forty-four thousand, four hundred and fifty-three) is:
:$11^3 \times 2663$
:The $9$th palindromic triangular number after $0$, $1$, $3$, $6$, $66$, $3003$, $66 \, 066$, $617 \, 716$ whose index is itself palindromic:
::$3 \, 544 \, 453 = T_{2662}$
:The $19$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$, $666$, $3003$, $5995$, $8778$, $15 \, 051$, $66 \, 066$, $617 \, 716$, $828 \, 828$, $1 \, 269 \, 621$, $1 \, 680 \, 861$:
::$3 \, 544 \, 453 = \dfrac {2662 \times \paren {2662 + 1} } 2$
:The $2662$nd triangular number:
::$3 \, 544 \, 453 = \dfrac {2662 \times \paren {2662 + 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:3,544,453
3,608,528,850,368,400,786,036,725

== Number ==
$3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725$ is:
:$5^2 \times 59 \times 15 \, 923 \times 78 \, 167 \times 1 \, 965 \, 575 \, 915 \, 851$
:The largest polydivisible number
== Also see ==
* Polydivisible Number: $3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725$
== Sources ==
* : $3,608,528,850,368,400,786,036,725$
Category:Specific Numbers
Category:3,608,528,850,368,400,786,036,725
3,628,800

== Number ==
$3 \, 628 \, 800$ (three million, six hundred and twenty-eight thousand, eight hundred ) is:
:$2^8 \times 3^4 \times 5^2 \times 7$
:The $10$th factorial after $1$, $2$, $6$, $24$, $120$, $720$, $5040$, $40 \, 320$, $362 \, 880$:
::$3 \, 628 \, 800 = 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
:Apart from the trivial $0!$, $1!$ and $2!$, the only factorial that is the product of $2$ consecutive factorials:
::$10! = 6! \times 7!$
== Also see ==
* 
== Sources ==
* : $3,628,800$
* : $3,628,800$
Category:Specific Numbers
Category:3,628,800
3,719,048,256

== Number ==
$3 \, 719 \, 048 \, 256$ is:
:$2^6 \times 3^4 \times 7^2 \times 11^4$
:The $60 \, 984$th square number:
::$3 \, 719 \, 048 \, 256 = 60 \, 984 \times 60 \, 984$
:The square of the larger of both the $1$st and the $2$nd pandigital pairs, with $35 \, 172$ and $57 \, 321$, of integers each have squares which are themselves pandigital:
::$35 \, 172^2 = 1 \, 237 \, 069 \, 584$, $60 \, 984^2 = 3 \, 719 \, 048 \, 256$
::$57 \, 321^2 = 3 \, 285 \, 697 \, 041$, $60 \, 984^2 = 3 \, 719 \, 048 \, 256$
== Also see ==
* Pandigital Pairs whose Squares are Pandigital
== Sources ==
* : $57,321$
Category:Specific Numbers
Category:3,719,048,256
3,785,942,160

== Number ==
$3 \, 785 \, 942 \, 160$ is:
:$2^4 \times 3^3 \times 5 \times 7 \times 11 \times 13 \times 17 \times 103$
:The $2$nd of $4$ pandigital integers which are divisible by all the positive integers up to $18$
== Also see ==
* 
== Sources ==
* : $2,438,195,760$
* : $2,438,195,760$
Category:Specific Numbers
Category:3,785,942,160
3-Digit Numbers forming Longest Reverse-and-Add Sequence

== Theorem ==
Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation.
Let $r \left({m}\right)$ be the reverse-and-add process on $m$.
Let $r$ be applied iteratively to $m$.
The $3$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are:
:$187, 286, 385, 583, 682, 781, 880$
all of which need $23$ iterations.
:$869$ and $968$ are the result of the first iteration for each of these, and so take $22$ iterations to reach a palindromic number.
== Proof ==
The sequence obtained by iterating $r$ on $187$ is:
:$187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688,$
:$17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933,$
:$47267087164, 93445163438, 176881317877, 955594506548, 170120002107, 8713200023178$
Note that $r \left({187}\right) = r \left({781}\right) = 968$, so the sequence obtained by iterating $r$ on $781$ is the same.
The rest of the numbers converge on the same sequence immediately:
:$286, 968, \ldots$
:$385, 968, \ldots$
:$880, 88, \ldots$
:$869, 1837, \ldots$
as do their reversals.
== Historical Note ==
== Sources ==
* : $187$
* : $196$
* : $187$
* : $196$
Category:Reverse-and-Add
3-Digit Numbers forming Longest Reverse-and-Add Sequence/Historical Note

== Historical Note on 3-Digit Numbers forming Longest Reverse-and-Add Sequence ==
 includes the following quote in his of $1997$:
:The remaining $75$ numbers can be classified into just a few groups, the members of which after one or two reversals each produce the same number and are therefore essentially the same. One of these groups consists of the numbers $187$, $286$, $385$, $583$, $682$, $781$, $869$, $880$ and $968$, each of which when reversed once or twice forms $1837$ and eventually forms the palindromic number $8,813,200,023,188$ after $23$ reversals.
He attributes this quote to Richard Hamilton, but no reference to it can be found on the internet, apart from its appearance in the above source work, and it has not been possible to corroborate it.
Technically speaking, of course, $869$ and $968$ take just $22$ iterations.
== Sources ==
* : $196$
* : $196$
Category:Historical Notes
3-Digit Numbers forming Longest Reverse-and-Add Sequence/Mistake

== Source Work ==
:
:The Dictionary
::$187$
:
:The Dictionary
::$187$
== Mistake ==
:The smallest of a group of $3$-digit numbers that require $23$ reversals to form a palindrome.
Incomplete and confusing.
What is being referred to here is the number of iterations of the reverse-and-add process.
== Sources ==
* : $187$
* : $187$
Category:Mistakes/Curious and Interesting Numbers
3-Digit Permutable Primes

== Theorem ==
The $3$-digit permutable primes are:
:$311, 199, 337$
and their anagrams, and no other.
== Proof ==
It is confirmed that:
:$113, 131, 311$ are all prime
:$199, 919, 991$ are all prime
:$337, 373, 733$ are all prime.
From Digits of Permutable Prime, all permutable primes contain digits in the set:
:$\left\{ {1, 3, 7, 9}\right\}$
The sum of a $3$-digit repdigit number is divisible by $3$.
By Divisibility by 3 it follows that all $3$-digit repdigit numbers are divisible by $3$ and therefore composite.
Hence the possibly permutable primes are:
:$113, 117, 119, 133, 137, 139, 177, 179, 199, 337, 339, 377, 379, 399, 779, 799$
and their anagrams.
We eliminate $113, 199, 337$ from this list, as it has been established that they are permutable primes.
Of those remaining, the following are composite:
It remains to demonstrate that at least one anagram of the remaining numbers:
:$137, 139, 179, 379$
is composite.
We find that:
All contenders are eliminated except for the established permutable primes $113, 119, 337$ and their anagrams.
== Sources ==
* : $113$
* : $113$
Category:Permutable Primes
3/Historical Note

== Historical Note on $3$ (three) ==
The number $3$ was considered by the ancient Greeks to be the first odd number, as they did not consider $1$ (one) a number, as such.
They associated the number $3$ with the triangle, with its $3$ vertices and $3$ sides.
To the Pythagoreans, odd and even numbers were considered to be either male or female, but sources differ on which was which.
Some suggest that $3$ was considered to be the first male number, being composed of unity ($1$) and $2$, the principle of diversity.
Such sources state that in contrast, the even numbers were considered to be female.
However, other sources suggest that it was the odd numbers which were female, while the even numbers were male.
In addition to that, in the eyes of the Pythagoreans, $3$ was in fact the first number, as in addition they considered that $2$ was not a number either, as it had a beginning and an end, but no middle.
 similarly considered $3$ to be the first number, but his reason was that it was the first number to be increased more by multiplication than by addition: $3 \times 3$ is greater than $3 + 3$.
$3$ is a common number into which to divide a body into parts.
For example:
:The positive, comparative and superlative of natural language.
:The world is divided into $3$ parts: the Underworld, the Earth (or Middle-Earth), and the Heavens.
:In the English language, the sequence (beloved of fairy tales) once, twice, thrice ends there -- there is no single word for "$n$ times" for any higher number.
In many cultures in history, $3$ is particularly significant.
In Greek mythology, there were:
:$3$ Fates
:$3$ Furies
:$3$ Graces
:$3 \times 3$ Muses
:Paris had to choose between $3$ goddesses
Oaths are repeated $3$ times.
Saint Peter denied Christ $3$ times.
The Bellman states, in , that:
:What I tell you three times is true.
== Sources ==
* : Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory
* : $3$
* : Chapter $\text {A}.2$: Pythagoras (ca. B.C.)
* : $3$
* : Chapter $2$: The Logic of Shape: Pythagoras
Category:Historical Notes
30,029

== Number ==
$30 \, 029$ (thirty thousand and twenty-nine) is:
:The $3248$th prime number
:The $4$th primorial prime after $5$, $29$, $2309$:
::$30 \, 029 = p_6 \# - 1 = 13 \# - 1 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 - 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:30,029
30,030

== Number ==
$30 \, 030$ (thirty thousand and thirty) is:
:$2 \times 3 \times 5 \times 7 \times 11 \times 13$
:The $6$th primorial after $1$, $2$, $6$, $30$, $210$, $2310$ (counting $1$ as the zeroth):
::$30 \, 030 = p_6 \# = 13 \# = 2 \times 3 \times 5 \times 7 \times 11 \times 13$
:Hence the smallest positive integer with $6$ distinct prime factors.
== Also see ==
* 
Category:Specific Numbers
Category:30,030
30,031

== Number ==
$30 \, 031$ (thirty thousand and thirty-one) is:
:$59 \times 509$
:The $1$st Euclid number which is composite:
::$30 \, 031 = p_6\# + 1 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1 = 59 \times 509$
:The $6$th Euclid number after $3$, $7$, $31$, $211$, $2311$:
::$30 \, 031 = p_6\# + 1 = 2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1$
== Also see ==
* 
* 
== Sources ==
* : $30,031$
Category:Specific Numbers
Category:30,031
30,240

== Number ==
$30 \, 240$ (thirty thousand, two hundred and forty) is:
:$2^5 \times 3^3 \times 5 \times 7$
:The $1$st quadruply perfect number:
::$\map {\sigma_1} {30 \, 240} = 120 \, 960 = 4 \times 30 \, 240$
== Also see ==
* 
== Sources ==
* : $30,240$
Category:Specific Numbers
Category:30,240
30,416,261,403

== Number ==
$30 \, 416 \, 261 \, 403$ is:
:$3 \times 11 \times 37 \times 101 \times 246 \, 643$
:The $11$th palindromic triangular number after $0$, $1$, $3$, $6$, $66$, $3003$, $66 \, 066$, $617 \, 716$, $3 \, 544 \, 453$, $6 \, 172 \, 882 \, 716$ whose index is itself palindromic:
::$30 \, 416 \, 261 \, 403 = T_{246 \, 642}$
:The $246 \, 642$nd triangular number:
::$30 \, 416 \, 261 \, 403 = \dfrac {246 \, 642 \times \paren {246 \, 642 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:30,416,261,403
30,739

== Number ==
$30 \, 739$ (thirty thousand, seven hundred and thirty-nine) is:
:$59 \times 521$
:Apart from $6$th powers, the value of $n$ less than $50 \, 000$ for which the difference between the fractional parts of $\sqrt n$ and $\sqrt [3] n$ is smallest
== Also see ==
* Smallest Differences between Fractional Parts of Square and Cube Roots
== Sources ==
* : $30,739$
* : $30,739$
Category:Specific Numbers
Category:30,739
300

== Number ==
$300$ (three hundred) is:
:$2^2 \times 3 \times 5^2$
:The $24$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $171$, $190$, $210$, $231$, $253$, $276$:
::$300 = \ds \sum_{k \mathop = 1}^{24} k = \dfrac {24 \times \paren {24 + 1} } 2$
:The $31$st highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $120$, $144$, $168$, $180$, $210$, $216$, $240$, $288$:
::$\map {\sigma_1} {300} = 868$
== Also see ==
* 
* 
Category:Specific Numbers
Category:300
3000/Mistake

== Source Work ==
:
:The Dictionary
::$3000$
== Mistake ==
:The smallest number requiring more than $12$ letters to write in the English language. It uses $13$.
Well, $101$ uses more than $12$ letters for a start:
:one hundred and one, for $16$ letters
or, in the American style:
:one hundred one, for $13$ letters.
== Sources ==
* : $3000$
Category:Mistakes/Curious and Interesting Numbers
3003

== Number ==
$3003$ (three thousand and three) is:
:$3 \times 7 \times 11 \times 13$
:The only known integer to appear as many as $8$ times in Pascal's triangle:
::$3003 = \dbinom {3003} 1 = \dbinom {78} 2 = \dbinom {15} 5 = \dbinom {14} 6 = \dbinom {14} 8 = \dbinom {15} {10} = \dbinom {78} {76} = \dbinom {3003} {3002}$
:The $6$th palindromic triangular number after $0$, $1$, $3$, $6$, $66$ whose index is itself palindromic:
::$3003 = T_{77}$
:The $10$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$, $666$.
:The $39$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $2145$, $2278$, $2415$, $2556$, $2701$, $2850$:
::$3003 = \ds \sum_{k \mathop = 1}^{39} \paren {4 k - 3} = 39 \paren {2 \times 39 - 1}$
:The $77$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2211$, $2278$, $2346$, $2415$, $2485$, $2556$, $2628$, $2701$, $2775$, $2850$, $2926$:
::$3003 = \ds \sum_{k \mathop = 1}^{77} k = \dfrac {77 \times \paren {77 + 1} } 2$
== Also see ==
* Numbers Appearing 8 Times in Pascal's Triangle
* 
* 
* 
* 
== Sources ==
* : $3003$
* : $3003$
Category:Specific Numbers
Category:3003
3008

== Number ==
$3008$ (three thousand and eight) is:
:$2^6 \times 47$
:The $32$nd octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $1281$, $1408$, $1541$, $1680$, $1825$, $1976$, $2133$, $2296$, $2465$, $2640$, $2821$:
::$3008 = \ds \sum_{k \mathop = 1}^{32} \paren {6 k - 5} = 32 \paren {3 \times 32 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:3008
301

== Number ==
$301$ (three hundred and one) is:
:$7 \times 43$
:The $14$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$, $100$, $126$, $155$, $187$, $222$, $260$:
::$301 = \dfrac {14 \paren {3 \times 14 + 1} } 2$
:The $28$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $126$, $145$, $176$, $187$, $210$, $222$, $247$, $260$, $287$:
::$301 = \dfrac {14 \paren {3 \times 14 + 1} } 2$
:The $45$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$:
::$301 \to 3^2 + 0^2 + 1^2 = 9 + 0 + 1 = 10 \to 1^2 + 0^2 = 1$
== Sources ==
* 
* 
* 
Category:Specific Numbers
Category:301
3010

== Number ==
$3010$ (three thousand and ten) is:
:$2 \times 5 \times 3 \times 43$
:The $35$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $2205$, $2356$, $2512$, $2673$, $2839$:
::$3010 = \ds \sum_{k \mathop = 1}^{35} \paren {5 k - 4} = \dfrac {35 \paren {5 \times 35 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3010
3015

== Number ==
$3015$ (three thousand and fifteen) is:
:$3^2 \times 5 \times 67$
:The $45$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $1820$, $1926$, $2035$, $2147$, $2262$, $2380$, $2501$, $2625$, $2752$, $2882$:
::$3015 = \ds \sum_{k \mathop = 1}^{45} \paren {3 k - 2} = \dfrac {45 \paren {3 \times 45 - 1} } 2$
:The $89$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $2380$, $2420$, $2501$, $2542$, $2625$, $2667$, $2752$, $2795$, $2882$, $2926$:
::$3015 = \ds \sum_{k \mathop = 1}^{45} \paren {3 k - 2} = \dfrac {45 \paren {3 \times 45 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3015
302

== Number ==
$302$ (three hundred and two) is:
:$2 \times 151$
:The $49$th nontotient:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 302$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The $46$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$:
::$302 \to 3^2 + 0^2 + 2^2 = 9 + 0 + 4 = 13 \to 1^2 + 3^2 = 1 + 9 = 10 \to 1^2 + 0^2 = 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:302
3024

== Number ==
$3024$ (three thousand and twenty-four) is:
:$2^4 \times 3^3 \times 7$
:The $13$th positive integer after $504$, $756$, $806$, $1008$, $1148$, $1209$, $1472$, $1512$, $2016$, $2208$, $2418$, $2924$ 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:
::$3024 = 36 \times 84 = 48 \times 63$
== Also see ==
* 
== Sources ==
* : $504$
* : $504$
Category:Specific Numbers
Category:3024
3025

== Number ==
$3025$ (three thousand and twenty-five) is:
:$5^2 \times 11^2$
:The sum of the first $10$ cubes:
::$3025 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 + 10^3$
:The $55$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1936$, $2025$, $2116$, $2209$, $2304$, $2401$, $2500$, $2601$, $2704$, $2809$, $2916$:
::$3025 = 55 \times 55$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3025
303

== Number ==
$303$ (three hundred and three) is:
:$3 \times 101$
:The $11$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $141$, $151$, $171$, $303$, $\ldots$
:The $56$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $237$, $241$, $259$, $261$, $267$, $273$, $283$, $285$, $289$, $297$, $303$, $\ldots$
== Also see ==
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::No further elements of this sequence will be tabulated on .
Category:Specific Numbers
Category:303
304

== Number ==
$304$ (three hundred and four) is:
:$2^4 \times 19$
:The $6$th primitive abundant number after $20$, $70$, $88$, $104$, $272$:
::$1 + 2 + 4 + 8 + 16 + 19 + 38 + 76 + 152 = 316 > 304$
:The $7$th primitive semiperfect number after $6$, $20$, $28$, $88$, $104$, $272$:
::$304 = 1 + 2 + 16 + 19 + 38 + 76 + 152$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $18$ different ways
:The $23$rd untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $262$, $268$, $276$, $290$, $292$
:The $50$th nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 304$
:where $\map \phi m$ denotes the Euler $\phi$ function
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Category:Specific Numbers
Category:304
305

== Number ==
$305$ (three hundred and five) is:
:$5 \times 61$
:The $17$th integer after $7$, $13$, $19$, $35$, $38$, $41$, $57$, $65$, $70$, $125$, $130$, $190$, $205$, $223$, $253$, $285$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$305^2 = 93 \, 025$; $9 = 3^2$, $3025 = 55^2$
== Also see ==
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Category:Specific Numbers
Category:305
306

== Number ==
$306$ (three hundred and six) is:
:$2 \times 3^2 \times 17$
:The $24$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $262$, $268$, $276$, $290$, $292$, $304$
== Also see ==
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== Historical Note ==
Category:Specific Numbers
Category:306
306/Historical Note

== Historical Note on 306 ==
$306$ was the number selected more or less at random by when he wished to test a new technique for calculating roots using continued fractions.
He calculated its $7$th root to $2800$ digits.
It starts:
:$2 \cdotp 26518 \ldots$
== Sources ==
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* : $306$
* : $306$
Category:Historical Notes
3060

== Number ==
$3060$ (three thousand and sixty) is:
:$2^2 \times 3^2 \times 5 \times 17$
:The $15$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$, $495$, $715$, $1001$, $1365$, $1820$, $2380$:
::$3060 = \ds \sum_{k \mathop = 1}^{15} \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {15 \paren {15 + 1} \paren {15 + 2} \paren {15 + 3} } {24}$
:The $45$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $1962$, $2072$, $2185$, $2301$, $2420$, $2542$, $2667$, $2795$, $2926$:
::$3060 = \dfrac {45 \paren {3 \times 45 + 1} } 2$
:The $90$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $2380$, $2420$, $2501$, $2542$, $2625$, $2667$, $2752$, $2795$, $2882$, $2926$, $3015$:
::$3060 = \dfrac {45 \paren {3 \times 45 + 1} } 2$
== Also see ==
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Category:Specific Numbers
Category:3060