nyuuzyou/wikis huggingface dataset:
4,478,976

== Number ==
$4 \, 478 \, 976$ (four million, four hundred and seventy-eight thousand, nine hundred and seventy-six) is:
:$2^{11} \times 3^7$
:The smallest $r$ satisfying the Diophantine equation $p^p \times q^q = r^r$:
::$2 \, 985 \, 984^{2 \, 985 \, 984} \times 1 \, 679 \, 616^{1 \, 679 \, 616} = 4 \, 478 \, 976^{4 \, 478 \, 976}$
== Also see ==
* Smallest Solution to Equation $p^p \times q^q = r^r$
== Sources ==
* : $4,478,976$
* : $4,478,976$
Category:Specific Numbers
Category:4,478,976
4,490,639

== Number ==
$4 \, 490 \, 639$ (four million, four hundred and ninety thousand, six hundred and thirty-nine) is:
:The $315 \, 347$th prime number
:The $3$rd term of the smallest Cunningham chain of the first kind of length $7$:
::$1 \, 122 \, 659$, $2 \, 245 \, 319$, $4 \, 490 \, 639$, $8 \, 981 \, 279$, $17 \, 962 \, 559$, $35 \, 925 \, 119$, $71 \, 850 \, 239$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4,490,639
4,679,307,774

== Number ==
$4 \, 679 \, 307 \, 774$ is:
:$2 \times 3^2 \times 193 \times 1 \, 346 \, 951$
:The $32$nd pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $\ldots$, $88 \, 593 \, 477$, $146 \, 511 \, 208$, $472 \, 335 \, 975$, $534 \, 494 \, 836$, $912 \, 985 \, 153$:
== Also see ==
* 
== Sources ==
* : $4,679,307,774$
* : $4,679,307,774$
Category:Specific Numbers
Category:4,679,307,774
4,700,063,497

== Number ==
$4 \, 700 \, 063 \, 497$ is:
:$19 \times 47 \times 5 \, 263 \, 229$
:The smallest positive integer $n$ such that $2^n - 3$ is divisible by $n$
== Also see ==
* Smallest n for which 2^n-3 is Divisible by n
== Sources ==
* : $4,700,063,497$
Category:Specific Numbers
Category:4,700,063,497
4,729,494

== Number ==
$4 \, 729 \, 494$ (four million, seven hundred and twenty-nine thousand, four hundred and ninety-four) is:
:$2 \times 3 \times 7 \times 11 \times 29 \times 353$
:The coefficient in Pell's equation leading to the solution of difficult version of Archimedes' cattle problem:
::$t^2 - 4 \, 729 \, 494 u^2 = 1$
== Also see ==
* Archimedes' Cattle Problem: Difficult Version
== Sources ==
* : $4,729,494$
* : $4,729,494$
Category:Specific Numbers
Category:4,729,494
4,753,869,120

== Number ==
$4 \, 753 \, 869 \, 120$ is:
:$2^6 \times 3^2 \times 5 \times 7 \times 11 \times 13 \times 17 \times 97$
:The $3$rd 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:4,753,869,120
4,782,969

== Number ==
$4 \, 782 \, 969$ (four million, seven hundred and eighty-two thousand, nine hundred and sixty-nine) is:
:$3^{14}$
:The $7$th power of $9$ after $(1)$, $9$, $81$, $729$, $6561$, $59 \, 049$, $531 \, 441$:
::$4 \, 782 \, 969 = 9^7$
:The $9$th $7$th power after $1$, $128$, $2187$, $16 \, 384$, $78 \, 125$, $279 \, 936$, $823 \, 543$, $2 \, 097 \, 152$:
::$4 \, 782 \, 969 = 9 \times 9 \times 9 \times 9 \times 9 \times 9 \times 9$
:The $14$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$, $243$, $729$, $2187$, $6561$, $19 \, 683$, $59 \, 049$, $177 \, 147$, $531 \, 441$, $1 \, 594 \, 323$:
::$4 \, 782 \, 969 = 3^{14}$
:The $2187$th square number:
::$4 \, 782 \, 969 = 2187 \times 2187$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:4,782,969
4,826,809

== Number ==
$4 \, 826 \, 809$ (four million, eight hundred and twenty-six thousand, eight hundred and nine) is:
:$13^6$
:The $6$th power of $13$ after $(1)$, $13$, $169$, $2197$, $28 \, 561$, $371 \, 293$:
::$4 \, 826 \, 809 = 13^6$
:The $13$th sixth power after $1$, $64$, $729$, $4096$, $15 \, 625$, $46 \, 656$, $117 \, 649$, $262 \, 144$, $531 \, 441$, $1 \, 000 \, 000$, $1 \, 771 \, 561$, $2 \, 985 \, 984$:
::$4 \, 826 \, 809 = 13 \times 13 \times 13 \times 13 \times 13 \times 13$
:The $169$th cube number:
::$4 \, 826 \, 809 = 169 \times 169 \times 169$
:The $2197$th square number:
::$4 \, 826 \, 809 = 2197 \times 2197$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:4,826,809
4,876,391,520

== Number ==
$4 \, 876 \, 391 \, 520$ is:
:$2^5 \times 3^2 \times 5 \times 7 \times 11 \times 13 \times 17 \times 199$
:The $4$th 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:4,876,391,520
4,937,775

== Number ==
$4 \, 937 \, 775$ (four million, nine hundred and thirty-seven thousand, seven hundred and seventy-five) is:
:$3 \times 5^2 \times 65 \, 837$
:The first Smith number to be discovered:
::$4 + 9 + 3 + 7 + 7 + 7 + 5 = 3 + 5 + 5 + 6 + 5 + 8 + 3 + 7 = 42$
== Also see ==
* Historical Note on Smith Numbers
== Sources ==
* : $4,937,775$
* : $4,937,775$
Category:Smith Numbers/Examples
Category:Specific Numbers
Category:4,937,775
4-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 $4$-digit integers $m$ which need the largest number of iterations before reaching a palindromic number are:
:$6999, 7998, 8997, 9996$
all of which need $20$ iterations.
== Proof ==
which is palindromic.
$7998$ and its reversal converge on the same sequence immediately:
== Sources ==
* : $6999$
* : $6999$
Category:Reverse-and-Add
4/Historical Note

== Historical Note on $4$ ==
The number $4$, as was $8$, was associated by the Pythagoreans with the concept of justice, being evenly balanced: $4 = 2 + 2$, where $2$ is the principle of diversity.
Throughout history, the number $4$ has been regarded with particular significance.
There were originally believed to be $4$ elements out of which everything was formed:
:Earth, Air, Fire and Water.
There are $4$ humours:
:Sanguine, Melacholic, Choleric and Phlegmatic.
There are $4$ cardinal points of the compass:
:North, East, South and West.
There are $4$ seasons of the year:
:Spring, Summer, Autumn and Winter.
In the Old Testament, there were $4$ rivers which watered the Garden of Eden:
:Pishon, Gihon, the Tigris, and the Euphrates.
In Islam, these rivers are:
:Saihan (Syr Darya), Jaihan (Amu Darya), Furat (Euphrates) and Nil (Nile).
In the New Testament, there are $4$ Gospels:
:Matthew, Mark, Luke and John.
To bring it all full circle, there are believed to be $4$ fundamental forces of nature:
:Gravitation, the Weak Force, the Electromagnetic Force, the Strong Force.
The most aesthetically pleasing musical intervals are those whose frequencies are associated with the ratio of $1 : 4$.
::Four strong winds that blow lonely
::Seven seas that run high
::All those things that don't change come what may
::But our good times are all gone
::And I'm bound for moving on
::I'll look for you if I'm ever back this way
:::: by 
== Sources ==
* : Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory
* : $4$
* : Chapter $\text {A}.2$: Pythagoras (ca. B.C.)
* : $4$
* : Chapter $2$: The Logic of Shape: Pythagoras
Category:Historical Notes
40

== Number ==
$40$ (forty) is:
:$2^3 \times 5$
:The only number whose name in the English language has all its letters in alphabetical order:
::forty
:The $4$th octagonal number, after $1$, $8$, $21$:
::$40 = 1 + 7 + 13 + 19 = 4 \paren {3 \times 4 - 2}$
:The $4$th pentagonal pyramidal number after $1$, $6$, $18$:
::$40 = 1 + 5 + 12 + 22 = \dfrac {4^2 \paren {4 + 1} } 2$
:The $5$th second pentagonal number after $2$, $7$, $15$, $26$:
::$40 = \dfrac {5 \paren {3 \times 5 + 1} } 2$
:The $7$th abundant number after $12, 18, 20, 24, 30, 36$:
::$1 + 2 + 4 + 5 + 8 + 10 + 20 = 50 > 40$
:The $9$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$:
::$40 = 2 + 8 + 10 + 20$
:The $10$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$:
::$40 = \dfrac {5 \paren {3 \times 5 + 1} } 2$
:The $16$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $\ldots$
:The $19$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $18$, $20$, $21$, $24$, $27$, $30$, $36$:
::$40 = 10 \times 4 = 10 \times \paren {4 + 0}$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
== Historical Note ==
== Sources ==
* : $40$
Category:Specific Numbers
Category:40
40,311

== Number ==
$40 \, 311$ (forty thousand, three hundred and eleven) is:
:$3^3 \times 1493$
:The $1$st of a sequence of $5$ consecutive integers which all have the same number of divisors:
::$\map {\sigma_0} {40 \, 311} = 8$: $1, 3, 9, 27, 1493, 4479, 13 \, 437, 40 \, 311$
== Also see ==
* 
== Sources ==
* : $40,311$
* : $40,311$
Category:Specific Numbers
Category:40,311
40,312

== Number ==
$40 \, 312$ (forty thousand, three hundred and twelve) is:
:$2^3 \times 5039$
:The $2$nd of a sequence of $5$ consecutive integers which all have the same number of divisors:
::$\map {\sigma_0} {40 \, 312} = 8$: $1, 2, 4, 8, 5039, 10 \, 078, 20 \, 156, 40 \, 312$
== Also see ==
* 
Category:Specific Numbers
Category:40,312
40,313

== Number ==
$40 \, 313$ (forty thousand, three hundred and thirteen) is:
:$7 \times 13 \times 443$
:The $3$rd of a sequence of $5$ consecutive integers which all have the same number of divisors:
::$\map {\sigma_0} {40 \, 313} = 8$: $1, 7, 13, 91, 443, 3101, 5759, 40 \, 313$
== Also see ==
* 
Category:Specific Numbers
Category:40,313
40,314

== Number ==
$40 \, 314$ (forty thousand, three hundred and fourteen) is:
:$2 \times 3 \times 6719$
:The $4$th of a sequence of $5$ consecutive integers which all have the same number of divisors:
::$\map {\sigma_0} {40 \, 314} = 8$: $1, 2, 3, 6, 6719, 13 \, 438, 20 \, 157, 40 \, 314$
== Also see ==
* 
Category:Specific Numbers
Category:40,314
40,315

== Number ==
$40 \, 315$ (forty thousand, three hundred and fifteen) is:
:$5 \times 11 \times 733$
:The $5$th of a sequence of $5$ consecutive integers which all have the same number of divisors:
::$\map {\sigma_0} {40 \, 315} = 8$: $1, 5, 11, 55, 733, 3665, 8063, 40 \, 315$
== Also see ==
* 
Category:Specific Numbers
Category:40,315
40,320

== Number ==
$40 \, 320$ (forty thousand, three hundred and twenty) is:
:$2^7 \times 3^2 \times 5 \times 7$
:The $8$th factorial after $1$, $2$, $6$, $24$, $120$, $720$, $5040$:
::$40 \, 320 = 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
== Also see ==
* 
== Sources ==
* : $40,320$
* : $40,320$
Category:Specific Numbers
Category:40,320
40,585

== Number ==
$40 \, 585$ (forty thousand, five hundred and eighty-five) is:
:$5 \times 8117$
:The $4$th and last factorion base $10$ after $1$, $2$, $145$:
::$40 \, 585 = 4! + 0! + 5! + 8! + 5!$
== Also see ==
* 
== Historical Note ==
== Sources ==
* : $145$
* : $40,585$
* : $145$
* : $40,585$
Category:Specific Numbers
Category:40,585
40,755

== Number ==
$40 \, 755$ (forty thousand, seven hundred and fifty-five) is:
:$3 \times 5 \times 11 \times 13 \times 19$
:The $2$nd integer after $1$ to be both pentagonal and hexagonal
:The $3$rd integer after $1$, $210$ to be both pentagonal and triangular
:The $143$rd hexagonal number:
::$40 \, 755 = \ds \sum_{k \mathop = 1}^{143} \paren {4 k - 3} = 143 \paren {2 \times 143 - 1}$
:The $165$th pentagonal number:
::$40 \, 755 = \ds \sum_{k \mathop = 1}^{165} \paren {3 k - 2} = \dfrac {165 \paren {3 \times 165 - 1} } 2$
:The $285$th triangular number:
::$40 \, 755 = \ds \sum_{k \mathop = 1}^{285} k = \dfrac {285 \times \paren {285 + 1} } 2$
:The $329$th generalized pentagonal number:
::$40 \, 755 = \ds \sum_{k \mathop = 1}^{165} \paren {3 k - 2} = \dfrac {165 \paren {3 \times 165 - 1} } 2$
== Also see ==
* 
* 
* 
* 
* 
* 
== Sources ==
* : $40,755$
* : $40,755$
Category:Specific Numbers
Category:40,755
40/Historical Note

== Historical Note on $40$ (Forty) ==
In the Bible, $40$ was frequently used as a number associated with a long period of time:
:The Israelites wandered for $40$ years in the Wilderness.
:Also in the Wilderness, Jesus fasted for $40$ days and $40$ nights.
There are $40$ rods, poles or perches in a furlong.
== Sources ==
* : $40$
* : $40$
Category:Historical Notes
400

== Number ==
$400$ (four hundred) is:
:$2^4 \times 5^2$
:The number of different ways of playing the first $2$ moves in chess
:The $2$nd after $121$ of the two square numbers which is the sum of consecutive powers of a positive integer:
::$400 = 7^0 + 7^1 + 7^2 + 7^3$
:The $6$th even integer after $2$, $4$, $94$, $96$, $98$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
:The $8$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$ to be the divisor sum of some (strictly) positive integer:
::$400 = \map {\sigma_1} {343}$
:The $18$th positive integer after $64$, $96$, $128$, $144$, $\ldots$, $320$, $324$, $336$, $352$, $360$, $384$ with $6$ or more prime factors:
::$400 = 2 \times 2 \times 2 \times 2 \times 5 \times 5$
:The $18$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $289$, $324$, $370$, $400$, $\ldots$
:The $20$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $225$, $256$, $289$, $324$, $361$:
::$400 = 20 \times 20$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $21$ different ways
:The $24$th number, and $3$rd square number after $1$, $81$, whose Divisor Sum is square:
:
:The $34$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $144$, $169$, $196$, $200$, $216$, $225$, $243$, $256$, $288$, $289$, $324$, $343$, $361$, $392$:
::$400 = 2^4 \times 5^2$
:The divisor sum of $7^3$, and so:
::$400 = 20^2 = 7^0 + 7^1 + 7^2 + 7^3$
== Also see ==
* Divisor Sum of 343
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $400$
* : $400$
Category:Specific Numbers
Category:400
4005

== Number ==
$4005$ (four thousand and five) is:
:$3^2 \times 5 \times 89$
:The $45$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $2145$, $2278$, $2415$, $2556$, $2701$, $2850$, $3003$, $3160$, $3321$, $3486$, $3655$, $3828$:
::$4005 = \ds \sum_{k \mathop = 1}^{45} \paren {4 k - 3} = 45 \paren {2 \times 45 - 1}$
:The $89$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $3240$, $3321$, $3403$, $3486$, $3570$, $3655$, $3741$, $3828$, $3916$:
::$4005 = \ds \sum_{k \mathop = 1}^{89} k = \dfrac {89 \times \paren {89 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4005
401

== Number ==
$401$ (four hundred and one) is:
:The $79$th prime number
:The smaller of the $4$th pair of primes whose prime gap is $8$:
::$409 - 401 = 8$
:The $8$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$:
::$401 = 20^2 + 1$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:401
4010

== Number ==
$4010$ (four thousand and ten) is:
:$2 \times 5 \times 401$
:The magic constant of a magic square of order $20$, after $1$, $(5)$, $15$, $34$, $\ldots$, $870$, $1105$, $1379$, $1695$, $2056$, $2465$, $2925$, $3439$:
::$\ds 4010 = \dfrac 1 {20} \sum_{k \mathop = 1}^{20^2} k = \dfrac {20 \paren {20^2 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:4010
402

== Number ==
$402$ (four hundred and two) is:
:$2 \times 3 \times 67$
:The $7$th even integer after $2$, $4$, $94$, $96$, $98$, $400$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
:The $44$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $290$, $310$, $318$, $322$, $345$, $354$, $357$, $366$, $370$, $374$, $385$, $399$:
::$402 = 2 \times 3 \times 67$
== Also see ==
* 
* 
Category:Specific Numbers
Category:402
403

== Number ==
$403$ (four hundred and three) is:
:$13 \times 31$
:The $13$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$, $342$:
::$403 = 1 + 7 + 11 + 16 + 21 + 26 + 31 + 36 + 41 + 46 + 51 + 56 = \dfrac {13 \paren {5 \times 13 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:403
4030

== Number ==
$4030$ (four thousand and thirty) is:
:$2 \times 5 \times 13 \times 31$
:The $3$rd weird number after $70$, $836$:
::$\map {\sigma_1} {4030} - 4030 = 4034$: its aliquot parts are $1$, $2$, $5$, $10$, $13$, $26$, $31$, $62$, $65$, $130$, $155$, $310$, $403$, $806$, $2015$, from which $4030$ cannot be made.
== Also see ==
* 
Category:Specific Numbers
Category:4030
4033

== Number ==
$4033$ (four thousand and thirty-three) is:
:$37 \times 109$
:The $13$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$, $2701$, $2821$, $3277$:
::$2^{4033} \equiv 2 \pmod {4033}$: $4033 = 37 \times 109$
:The $37$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $2296$, $2465$, $2640$, $2821$, $3008$, $3201$, $3400$, $3605$, $3816$:
::$4033 = \ds \sum_{k \mathop = 1}^{37} \paren {6 k - 5} = 37 \paren {3 \times 37 - 2}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:4033
404

== Number ==
$404$ (four hundred and four) is:
:$2^2 \times 101$
:The $8$th even integer after $2$, $4$, $94$, $96$, $98$, $400$, $402$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
:The $41$st noncototient after $10$, $26$, $34$, $50$, $\ldots$, $344$, $346$, $362$, $366$, $372$, $386$, $394$:
::$\nexists m \in \Z_{>0}: m - \map \phi n = 404$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:404
405

== Number ==
$405$ (four hundred and five) is:
:$3^4 \times 5$
:The $9$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $126$, $196$, $288$:
::$405 = 1 + 5 + 12 + 22 + 35 + 51 + 70 + 92 + 117 = \dfrac {9^2 \paren {9 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:405
4053

== Number ==
$4053$ (four thousand and fifty-three) is:
:$3 \times 7 \times 193$
:The $19$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$, $160$, $661$, $2653$, $3069$, $3943$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$ is prime
== Also see ==
* 
Category:Specific Numbers
Category:4053
406

== Number ==
$406$ (four hundred and six) is:
:$2 \times 7 \times 29$
:The $1$st term of the $1$st sequence of $3$ consecutive triangular numbers all of which are sphenic:
:The $28$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $171$, $190$, $210$, $231$, $253$, $276$, $300$, $325$, $351$, $378$:
::$406 = \ds \sum_{k \mathop = 1}^{28} k = \dfrac {28 \times \paren {28 + 1} } 2$
:The $31$st untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $324$, $326$, $336$, $342$, $372$
:The $45$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $290$, $310$, $318$, $322$, $345$, $354$, $357$, $366$, $370$, $374$, $385$, $399$, $402$:
::$406 = 2 \times 7 \times 29$
== Also see ==
* Sequence of Smallest 3 Consecutive Triangular Numbers which are Sphenic
* 
* 
* 
* 
Category:Specific Numbers
Category:406
4060

== Number ==
$4060$ (four thousand and sixty) is:
:$2^2 \times 5 \times 7 \times 29$
:The $28$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $816$, $969$, $1140$, $1330$, $1540$, $1771$, $2024$, $2300$, $2600$, $2925$, $3276$, $3276$:
::$4060 = \ds \sum_{k \mathop = 1}^{28} \frac {k \paren {k + 1} } 2 = \dfrac {28 \paren {28 + 1} \paren {28 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:4060
407

== Number ==
$407$ (four hundred and seven) is:
:$11 \times 37$
:The $13$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $153$, $370$, $371$:
::$4^3 + 0^3 + 7^3 = 64 + 0 + 343 = 407$
== Also see ==
* 
== Sources ==
* : $153$
* : $407$
* : $153$
* : $407$
Category:Specific Numbers
Category:407
408

== Number ==
$408$ (four hundred and eight) is:
:$2^3 \times 3 \times 17$
:The $12$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$, $225$, $280$, $341$:
::$408 = \ds \sum_{k \mathop = 1}^{12} \paren {6 k - 5} = 12 \paren {3 \times 12 - 2}$
:The $32$nd untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $324$, $326$, $336$, $342$, $372$, $406$
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Category:Specific Numbers
Category:408
409

== Number ==
$409$ (four hundred and nine) is:
:The $80$th prime number
:The $2$nd after $199$ of the smallest $10$ primes in arithmetic sequence:
::$409 = 1 \times 210 + 199$
:The larger of the $4$th pair of primes whose prime gap is $8$:
::$409 - 401 = 8$
:The smaller of the $6$th pair of primes whose prime gap is $10$:
::$419 - 409 = 10$
:The $10$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$
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Category:Specific Numbers
Category:409
409,113

== Number ==
$409 \, 113$ (four hundred and nine thousand, one hundred and thirteen) is:
:$3^2 \times 131 \times 347$
:The sum of the first $9$ factorials:
::$409 \, 113 = 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9!$
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Category:Specific Numbers
Category:409,113
4093

== Number ==
$4093$ (four thousand and ninety-three) is:
:The $564$th prime number
:The $13$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$, $337$, $991$, $1873$, $2053$, $2377$, $4093$
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Category:4093
4095

== Number ==
$4095$ (four thousand and ninety-five) is:
:$3^2 \times 5 \times 7 \times 13$
:The $5$th and last Ramanujan-Nagell number after $0$, $1$, $3$, $15$:
::$4095 = 2^{12} - 1 = \dfrac {90 \times \paren {90 + 1} } 2$
:The $6$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$:
::$\map {\sigma_1} {4095} - 4095 = 4641 > 4095$
:The $90$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $3655$, $3741$, $3828$, $3916$, $4005$:
::$4095 = \ds \sum_{k \mathop = 1}^{90} k = \dfrac {90 \times \paren {90 + 1} } 2$
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Category:4095
4096

== Number ==
$4096$ (four thousand and ninety-six) is:
:$2^{12}$
:Hence in binary notation:
::$1 \, 000 \, 000 \, 000 \, 000$
:The $3$rd power of $16$ after $(1)$, $16$, $256$:
::$4096 = 16^3$
:The $4$th $6$th power after $1$, $64$, $729$:
::$4096 = 4 \times 4 \times 4 \times 4 \times 4 \times 4$
:The $4$th power of $8$ after $(1)$, $8$, $64$, $512$:
::$4096 = 8^4$
:The $6$th power of $4$ after $(1)$, $4$, $16$, $64$, $256$, $1024$:
::$4096 = 4^6$
:The $8$th fourth power after $1$, $16$, $81$, $256$, $625$, $1296$, $2401$:
::$4096 = 8 \times 8 \times 8 \times 8$
:Hence in octal notation:
::$10 \, 000$
:The $12$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$:
::$4096 = 2^{12}$
:The $13$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$:
::$\map {\sigma_1} {4096} = 8191 = 2 \times 4096 - 1$
:The $16$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$, $1728$, $2197$, $2744$, $3375$:
::$4096 = 16 \times 16 \times 16$
:Hence in hexadecimal notation:
::$1000$
:The $24$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $\ldots$, $1936$, $2304$, $2704$, $2916$, $3136$, $3600$, $3844$ to be the divisor sum value of some (strictly) positive integer:
::$4096 = \map {\sigma_1} {2667} = \map {\sigma_1} {3937}$
:The $64$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $3249$, $3364$, $3481$, $3600$, $3721$, $3844$, $3969$:
::$4096 = 64 \times 64$
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== Sources ==
* : $4096$
* : $4096$
Category:Specific Numbers
Category:4096