nyuuzyou/wikis huggingface dataset:
71

== Number ==
$71$ (seventy-one) is:
:The $20$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$, $61$, $67$
:The $2$nd prime number after $53$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$.
:The $3$rd prime number after $2, 5$ which divides the sum of all smaller primes:
::$8 \times 71 = 568 = 2 + 3 + 5 + \cdots + 61 + 67$
:The $5$th emirp after $13$, $17$, $31$, $37$
:The smaller of the $8$th pair of twin primes, with $73$
:The $10$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$
:The $11$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$, $53$, $59$, $67$ such that the Mersenne number $2^p - 1$ is composite
:The $11$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$
:Its square is the sum of two factorials:
::$71^2 = 7! + 1!$
:Its cube is the odd integers from $3$ to $11$ written in sequence:
::$71^3 = 357 \, 911$
== Also see ==
* Brocard's Problem
* Cube of 71 is Odd Integers in Sequence
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== Sources ==
* : $71$
* : $71$
Category:Specific Numbers
Category:71
71,850,239

== Number ==
$71 \, 850 \, 239$ (seventy-one million, eight hundred and fifty thousand, two hundred and thirty-nine) is:
:The $4 \, 220 \, 407$rd prime number
:The $7$th and last 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:71,850,239
71,973

== Number ==
$71 \, 973$ (seventy-one thousand, nine hundred and seventy-three) is:
:$3^2 \times 11 \times 727$
:One of the elements of one of the $4$-cycles to which Kaprekar's process on some $5$-digit numbers converges:
::$71 \, 973 \to 83 \, 952 \to 74 \, 943 \to 62 \, 964 \to 71 \, 973$
== Also see ==
* 
Category:Specific Numbers
Category:71,973
7105

== Number ==
$7105$ (seven thousand, one hundred and five) is:
:$5 \times 7^2 \times 29$
:The $49$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $5985$, $6256$, $6533$, $6816$:
::$7105 = \ds \sum_{k \mathop = 1}^{49} \paren {6 k - 5} = 49 \paren {3 \times 49 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:7105
714

== Number ==
$714$ (seven hundred and fourteen) is:
:$2 \times 3 \times 7 \times 17$
:The $1$st of the $5$th (and largest known) pair of consecutive integers whose product is a primorial:
::$714 \times 715 = 510 \, 510 = 17 \#$
:The $8$th integer after $1$, $14$, $30$, $105$, $248$, $264$, $418$ whose divisor sum divided by its Euler $\phi$ value is a square:
::$\dfrac {\map {\sigma_1} {714} } {\map \phi {714} } = \dfrac {1728} {192} = 9 = 3^2$
:The $10$th positive integer after $1$, $7$, $102$, $110$, $142$, $159$, $187$, $381$, $690$ the sum of whose divisors is a cube:
::$\map {\sigma_1} {714} = 1728 = 12^3$
:The $21$st integer $n$ after $1$, $3$, $15$, $30$, $35$, $56$, $\ldots$, $210$, $248$, $264$, $357$, $420$, $570$, $616$, $630$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {714} = 16$, $\map \phi {714} = 192$, $\map {\sigma_1} {714} = 1728$
== Also see ==
* Integers whose Ratio between Divisor Sum and Phi is Square
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== Historical Note ==
== Sources ==
* : $714$
* : $714$
Category:Specific Numbers
Category:714
714/Historical Note

== Historical Note on $714$ (Seven Hundred and Fourteen) ==
:On April 8, 1974, in Atlanta, Georgia, hit his $715$th , thus eclipsing the previous mark of $714$ long held by . This event received so much advance publicity that the numbers $714$ and $715$ were on millions of lips. Questions like, 'When do you think he'll get $715$?' were perfectly understood, even with no mention made of Aaron, Ruth or homerun. In all the hubbub it appears certain interesting properties of $714$ and $715$ were overlooked ...
::::-- , and 
The whimsical paper so authored was noticed by . This led to a long and fruitful collaboration between and .
== Sources ==
* 
* : $714$
* : $714$
Category:Historical Notes
7140

== Number ==
$7140$ (seven thousand, one hundred and forty) is:
:$2^2 \times 3 \times 5 \times 7 \times 17$
:The $5$th and last after $1$, $10$, $120$, $1540$ of the $5$ tetrahedral numbers which are also triangular.
:The $34$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $\ldots$, $3276$, $3276$, $4060$, $4495$, $4960$, $5456$, $5984$, $6545$:
::$7140 = \ds \sum_{k \mathop = 1}^{34} \frac {k \paren {k + 1} } 2 = \dfrac {34 \paren {34 + 1} \paren {34 + 2} } 6$
:The $60$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $5565$, $5778$, $5995$, $6216$, $6441$, $6670$, $6903$:
::$7140 = \ds \sum_{k \mathop = 1}^{60} \paren {4 k - 3} = 60 \paren {2 \times 60 - 1}$
:The $119$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5995$, $6105$, $6216$, $6328$, $6441$, $6555$, $6670$, $6786$, $6903$, $7021$:
::$7140 = \ds \sum_{k \mathop = 1}^{119} k = \dfrac {119 \times \paren {119 + 1} } 2$
== Also see ==
* 
* 
* 
* 
== Sources ==
* : $7140$
* : $7140$
Category:Specific Numbers
Category:7140
715

== Number ==
$715$ (seven hundred and fifteen) is:
:$5 \times 11 \times 13$
:The $2$nd of the $5$th (and largest known) pair of consecutive integers whose product is a primorial:
::$714 \times 715 = 510 \, 510 = 17 \#$
:The $10$th pentatope number after $1$, $5$, $15$, $35$, $70$, $126$, $210$, $330$, $495$:
::$715 = \ds \sum_{k \mathop = 1}^{10} \dfrac {k \paren {k + 1} \paren {k + 2} } 6 = \dfrac {10 \paren {10 + 1} \paren {10 + 2} \paren {10 + 3} } {24}$
:The $22$nd pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$, $117$, $145$, $176$, $210$, $247$, $287$, $330$, $330$, $376$, $425$, $477$, $532$, $590$, $651$:
::$715 = \ds \sum_{k \mathop = 1}^{22} \paren {3 k - 2} = \dfrac {22 \paren {3 \times 22 - 1} } 2$
:The $43$rd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$, $551$, $590$, $610$, $651$, $672$:
::$715 = \ds \sum_{k \mathop = 1}^{22} \paren {3 k - 2} = \dfrac {22 \paren {3 \times 22 - 1} } 2$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:715
715,716

== Number ==
$715 \, 716$ (seven hundred and fifteen thousand, seven hundred and sixeen) is:
:$2^2 \times 3^4 \times 47^2$
:The $846$th square number:
::$715 \, 716 = 846 \times 846$
:The $5$th square number after $8281$, $183 \, 184$, $328 \, 329$, $528 \, 529$ whose digits form $2$ consecutive integers:
::$715 \, 716 = 846^2$
:The $4$th square number after $183 \, 184$, $328 \, 329$, $528 \, 529$ whose digits form $2$ consecutive increasing integers:
::$715 \, 716 = 846^2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:715,716
717

== Number ==
$717$ (seven hundred and seventeen) is:
:$3 \times 239$
:The $11$th integer after $0$, $1$, $3$, $5$, $7$, $9$, $33$, $99$, $313$, $585$ which is palindromic in both decimal and binary:
::$717_{10} = 1 \, 011 \, 001 \, 101_2$
:The $14$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $141$, $151$, $171$, $303$, $393$, $535$, $717$, $\ldots$
== Also see ==
* 
* 
Category:Specific Numbers
Category:717
719

== Number ==
$719$ (seven hundred and nineteen) is:
:The $128$th prime number.
:The $3$rd prime number after $5$, $23$ of the form $n! - 1$ for integer $n$:
::$719 = 6! - 1$
:where $n!$ denotes $n$ factorial
:The $4$th term after $89$, $179$, $359$ of the smallest Cunningham chain of the first kind of length $6$:
::$\tuple {89, 179, 359, 719, 1439, 2879}$
:The smaller of the $10$th pair of primes whose prime gap is $8$:
::$727 - 719 = 8$
:The larger of the $12$th pair of primes whose prime gap is $10$:
::$719 - 709 = 10$
:The $21$st safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$, $347$, $359$, $383$, $467$, $479$, $503$, $563$, $587$:
::$719 = 2 \times 359 + 1$
:The $24$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$, $233$, $239$, $293$, $311$, $313$, $317$, $373$, $379$, $593$, $599$
:The $32$nd Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $233$, $239$, $251$, $281$, $293$, $359$, $419$, $431$, $443$, $491$, $509$, $593$, $641$, $653$, $659$, $683$:
::$2 \times 719 + 1 = 1439$, which is prime.
== Also see ==
* 
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== Sources ==
* : $719$
Category:Specific Numbers
Category:719
7192

== Number ==
$7192$ (seven thousand, one hundred and ninety-two) is:
:$2^3 \times 29 \times 31$
:The $5$th weird number after $70$, $836$, $4030$, $5830$:
::$\map {\sigma_1} {7192} - 7192 = 7208$: $7192$ cannot be made from its aliquot parts
== Also see ==
* 
Category:Specific Numbers
Category:7192
72

== Number ==
$72$ (seventy-two) is:
:$2^3 \times 3^2$
:The $1$st element of the $1$st set of $4$ positive integers which form an arithmetic sequence which all have the same Euler $\phi$ value:
::$\map \phi {72} = \map \phi {78} = \map \phi {84} = \map \phi {90} = 24$
:The $1$st positive integer whose fifth power can be expressed as the sum of $5$ other fifth powers:
::$72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5$
:The $12$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$
:The $16$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$, $66$:
::$72 = 12 + 24 + 36$
:There are $17$ positive integers which have an Euler $\phi$ value $72$.
:The $18$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$:
::$\map {\sigma_1} {72} = 195$
:The $21$st Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$, $18$, $26$, $28$, $36$, $38$, $47$, $48$, $53$, $57$, $62$, $69$:
::$72 = 69 + 3$
:The $25$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $67$, $72$, $\ldots$
:The $32$nd integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{72} = 4 \, 722 \, 366 \, 482 \, 869 \, 645 \, 213 \, 696$
:The $40$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $50$, $54$, $55$, $59$, $60$, $61$, $65$, $66$, $67$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $42$nd (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $55$, $60$, $61$, $65$, $66$, $67$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The product of the number of edges, edges per face and faces of a tetrahedron.
::$6 \times 4 \times 3$
== Also see ==
* Numbers with Euler Phi Value of 72
* Product of Number of Edges, Edges per Face and Faces of Tetrahedron
* Smallest 5th Power equal to Sum of 5 other 5th Powers
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Category:Specific Numbers
Category:72
720

== Number ==
$720$ (seven hundred and twenty) is:
:$2^4 \times 3^2 \times 5$
:The $6$th factorial after $1$, $2$, $6$, $24$, $120$:
::$720 = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$
:The $14$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$:
::$\map {\sigma_0} {720} = 24$
:The $14$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$:
::$\dfrac {\map {\sigma_1} {720} } {720} = \dfrac {2418} {720} = 3 \cdotp 358 \dot 3$
:The $16$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$, $512$, $576$, $640$, $648$, $672$, $704$ with $7$ or more prime factors:
::$720 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5$
:The $41$st highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $210$, $216$, $240$, $288$, $300$, $336$, $360$, $420$, $480$, $504$, $600$, $630$, $660$:
::$\map {\sigma_1} {720} = 2418$
:The product of consecutive integers in $2$ different ways:
::$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$
:The only factorial which can be expressed as the sum of two squares:
::$6! = 12^2 + 24^2$
== Also see ==
* 720 is Product of Consecutive Numbers in Two Ways
* Factorial which is Sum of Two Squares
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== Sources ==
* : Glossary
* : $720$
* : Glossary
* : $720$
Category:Specific Numbers
Category:720
7200

== Number ==
$7200$ (seven thousand, two hundred) is:
:$2^5 \times 3^2 \times 5^2$
:The $24$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $\ldots$, $1470$, $1800$, $2176$, $2601$, $3078$, $3610$, $4200$, $4851$, $5566$, $6348$:
::$7200 = \ds \sum_{k \mathop = 1}^{24} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {24^2 \paren {24 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:7200
720 is Product of Consecutive Numbers in Two Ways

== Theorem ==
:$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$
== Proof ==
Follows from Factorial as Product of Two Factorials:
:$10! = 6! \times 7!$
and so:
:$\dfrac {10!} {7!} = 10 \times 9 \times 8 = 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Hence the result.
== Sources ==
* : $720$
* : $720$
Category:Factorials
Category:720
720 is Product of Consecutive Numbers in Two Ways/Mistake

== Source Work ==
:
:The Dictionary
::$720$
== Mistake ==
:$10! = 7! \times 6!$, the only known example of a factorial being the product of two other factorials.
== Correction ==
As is pointed out in Factorial as Product of Two Factorials, for all $n \in \Z_{>0}$:
:$\paren {n!}! = n! \paren {n! - 1}!$
So the above statement should be amended, for example as is done in the section on the number $10$, to say something like:
:Apart from the general rule ...
== Sources ==
* : $720$
Category:Mistakes/Curious and Interesting Numbers
721

== Number ==
$721$ (seven hundred and twenty-one) is:
:$7 \times 103$
:The $16$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$, $397$, $469$, $547$, $631$:
::$721 = \ds 1 + \sum_{k \mathop = 1}^{16 - 1} 6 k = 16^3 - 15^3$
== Also see ==
* 
Category:Specific Numbers
Category:721
7225

== Number ==
$7225$ (seven thousand, two hundred and twenty-five) is:
:$5^2 \times 17^2$
:The $85$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $6084$, $6241$, $6400$, $6561$, $6724$, $6889$, $7056$
::$7225 = 85 \times 85$
== Also see ==
* 
Category:Specific Numbers
Category:7225
7245

== Number ==
$7245$ (seven thousand, two hundred and forty-five) is:
:$3^2 \times 5 \times 7 \times 23$
:The $14$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$, $4725$, $5355$, $5775$, $5985$, $6435$, $6615$, $6825$:
::$\map {\sigma_1} {7245} - 7245 = 7731 > 7245$
== Also see ==
* 
Category:Specific Numbers
Category:7245
725

== Number ==
$725$ (seven hundred and twenty-five) is:
:$5^2 \times 29$
:The $4$th positive integer after $325$, $425$, $650$ which can be expressed as the sum of two square numbers in $3$ distinct ways:
::$725 = 26^2 + 7^2 = 25^2 + 10^2 = 23^3 + 14^2$
== Also see ==
* 
Category:Specific Numbers
Category:725
726

== Number ==
$726$ (seven hundred and twenty-six) is:
:$2 \times 3 \times 11^2$
:The $11$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$, $126$, $196$, $288$, $405$, $550$:
::$726 = \ds \sum_{k \mathop = 1}^{11} \dfrac {k \paren {3 k - 1} } 2 = \dfrac {11^2 \paren {11 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:726
7260

== Number ==
$7260$ (seven thousand, two hundred and sixty) is:
:$2^2 \times 3 \times 5 \times 11^2$
:The $120$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $6786$, $6903$, $7021$, $7140$:
::$7260 = \ds \sum_{k \mathop = 1}^{120} k = \dfrac {120 \times \paren {120 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:7260
727

== Number ==
$727$ (seven hundred and twenty-seven) is:
:The $129$th prime number
:The $3$rd integer after $428$, $573$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive increasing integers:
::$727^2 = 528 \, 529$
:The $4$th integer after $91$, $428$, $573$ whose square has a decimal representation consisting of the concatenation of $2$ consecutive integers:
::$727^2 = 528 \, 529$
:The larger of the $10$th pair of primes whose prime gap is $8$:
::$727 - 719 = 8$
:The $15$th palindromic prime:
::$2$, $3$, $5$, $7$, $11$, $101$, $131$, $151$, $181$, $191$, $313$, $353$, $373$, $383$, $727$, $\ldots$
:The $15$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $141$, $151$, $171$, $303$, $393$, $535$, $717$, $727$, $\ldots$
:The $30$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $353$, $373$, $383$, $433$, $443$, $449$, $499$, $557$, $577$, $599$, $661$, $677$
:The $47$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $491$, $499$, $503$, $509$, $541$, $571$, $577$, $593$, $619$, $647$, $659$, $701$, $709$
== Also see ==
* 
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Category:Specific Numbers
Category:727
7272

== Number ==
$7272$ (seven thousand, two hundred and seventy-two) is:
:$2^3 \times 3^2 \times 101$
:The $15$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$, $703$, $999$, $2223$, $2728$, $4879$, $4950$, $5050$, $5292$:
::$7272^2 = 52 \, 881 \, 984 \to 5288 + 1984 = 7272$
== Also see ==
* 
Category:Specific Numbers
Category:7272
728

== Number ==
$728$ (seven hundred and twenty-eight) is:
:$2^3 \times 7 \times 13$
:The $1$st positive integer which can be expressed as the sum of $2$ cubes in $3$ different ways:
::$728 = 6^3 + 8^3 = \paren {-1}^3 + 9^3 = \paren {-10}^3 + 12^3$
:The smaller of the $1$st pair of Smith brothers:
::$7 + 2 + 8 = 2 + 2 + 2 + 7 + 1 + 3 = 17$, $7 + 2 + 9 = 3 + 3 + 3 + 3 + 3 + 3 = 18$
:The $37$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$, $654$, $663$, $666$, $690$, $706$:
::$7 + 2 + 8 = 2 + 2 + 2 + 7 + 1 + 3 = 17$
== Also see ==
* 
* 
* 
== Sources ==
* : $728$
Category:Specific Numbers
Category:728
729

== Number ==
$729$ (seven hundred and twenty-nine) is:
:$3^6$
:$1 \, 000 \, 000$ in base $3$
:The $1$st solution to the approximate Fermat equation $x^3 = y^3 + z^3 \pm 1$:
::$9^3 = 6^3 + 8^3 + 1$
:The $1$st cube which can be expressed as the sum of $5$ positive cubes:
::$729 = 1^3 + 3^3 + 4^3 + 5^3 + 8^3$
:The larger of the $1$st pair of Smith brothers:
::$7 + 2 + 8 = 2 + 2 + 2 + 7 + 1 + 3 = 17$, $7 + 2 + 9 = 3 + 3 + 3 + 3 + 3 + 3 = 18$
:The $2$nd cube which can be expressed as the sum of $3$ positive cubes:
::$729 = 1^3 + 6^3 + 8^3$
:The $3$rd $6$th power after $1$, $64$:
::$729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3$
:The number of different commutative binary operations that can be applied to a set with $3$ elements
:The $3$rd power of $9$ after $(1)$, $9$, $81$:
::$729 = 9^3$
:The $6$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$, $243$:
::$729 = 3^6$
:The $9$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$:
::$729 = 9 \times 9 \times 9$
:The $27$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $361$, $400$, $441$, $484$, $529$, $576$, $625$, $676$:
::$729 = 27 \times 27$
:The $38$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$, $654$, $663$, $666$, $690$, $706$, $728$:
::$7 + 2 + 9 = 3 + 3 + 3 + 3 + 3 + 3 = 18$
:The $46$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $400$, $432$, $441$, $484$, $500$, $512$, $529$, $576$, $625$, $648$, $675$, $676$:
::$729 = 3^6$
== Also see ==
* Period of Reciprocal of 729 is 81
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== Historical Note ==
== Sources ==
* : $729$
* : $729$
Category:Specific Numbers
Category:729
729/Historical Note

== Historical Note on $729$ ==
$729$ was particularly significant to the Pythagoreans, as it is $27^2$ as well as being $9^3$.
The philosopher rejected the well-known length of the year as being (approximately) $365 \frac 1 4$ in favour of $364 \frac 1 4$, as the latter is $729$, that is $9^3$, days and nights.
This was on the grounds that:
:... if one were to express the extent of the interval by which the tyrant is parted from the king in respect of true pleasure he will find on completion of the multiplication that he lives $729$ times as happily and that the tyrant's life is more painful by the same distance.
::::-- 's : $588$
Various other interpretations of this passage have been suggested.
 combined the sequences of squares and cubes and their roots from $1$ to $3$ to get:
:$1 + 2 + 3 + 4 + 8 + 9 = 27$
 pointed out that the $27 \times 27$ magic square, being filled with the numbers from $1$ to $729$, has $365$ in the centre cell.
== Sources ==
* 
* : $729$
* : Chapter $\text {A}.2$: Pythagoras (ca. B.C.): Footnote $5$
* : $729$
Category:Historical Notes
73,939,133

== Number ==
$73 \, 939 \, 133$ is:
:The $4 \, 335 \, 890$th prime number
:The $83$rd and last right-truncatable prime:
::$7 \, 393 \, 913$, $739 \, 391$, $73 \, 939$, $7393$, $739$, $73$ and $7$ are all prime
== Also see ==
* 
* 
== Sources ==
* : $73,939,133$
Category:Specific Numbers
Category:73,939,133
73/Sheldon Cooper Exposition

== The Exposition on the Number $73$ ==
:
:$73$ is the best number:
::$73$ is the $21$st prime number.
::Its mirror $37$ is the $12$th prime number.
::Its mirror $21$ is the product of multiplying, hang on to your hats, $7$ by $3$.
:Eh? Eh? Did I lie?
:
:We get it. $73$ is the of numbers.
:
: wishes.
: In binary, $73$ is a palindrome: $1,001,001$, which backwards is $1,001,001$, exactly the same.
: All backwards gets you is Sirron Kcuhc!
::::, Episode $73$: Lee Aronsohn, Jim Reynolds and Maria Ferrari
== Sources ==
* 
Category:73
730

== Number ==
$730$ (seven hundred and thirty) is:
:$2 \times 5 \times 73$
:The $12$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $91$, $130$, $214$, $226$, $370$, $526$, $706$, $730$, $\ldots$
:The $24$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $289$, $324$, $370$, $400$, $526$, $529$, $625$, $676$, $706$, $730$, $\ldots$
== Also see ==
* 
* 
Category:Specific Numbers
Category:730
7314

== Number ==
$7314$ (seven thousand, three hundred and fourteen) is:
:$2 \times 3 \times 23 \times 53$
:The smaller of the $1$st pair of consecutive positive integers which are each the product of exactly $4$ distinct prime numbers:
::$7314 = 2 \times 3 \times 23 \times 53$, $7315 = 5 \times 7 \times 11 \times 19$
== Also see ==
* 
== Sources ==
* : $7314$
Category:Specific Numbers
Category:7314
7315

== Number ==
$7315$ (seven thousand, three hundred and fifteen) is:
:$5 \times 7 \times 11 \times 19$
:The larger of the $1$st pair of consecutive positive integers which are each the product of exactly $4$ distinct prime numbers:
::$7314 = 2 \times 3 \times 23 \times 53$, $7315 = 5 \times 7 \times 11 \times 19$
== Also see ==
* 
== Sources ==
* : $7314$
Category:Specific Numbers
Category:7315
733

== Number ==
$733$ (seven hundred and thirty-three) is:
:The $130$th prime number
:The $13$th balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$, $563$, $593$, $607$, $653$:
::$733 = \dfrac {727 + 739} 2$
:The $20$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $113$, $131$, $199$, $311$, $337$, $373$
:The $23$rd emirp after $13$, $17$, $31$, $37$, $\ldots$, $311$, $337$, $347$, $359$, $389$, $709$
:The $25$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $\ldots$, $317$, $373$, $379$, $593$, $599$, $719$
:The $31$st near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $353$, $373$, $383$, $433$, $443$, $449$, $499$, $557$, $577$, $599$, $661$, $677$, $727$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:733