nyuuzyou/wikis huggingface dataset:
6561

== Number ==
$6561$ (six thousand, five hundred and sixty-one) is:
:$3^8$
:The $3$rd positive integer after $1$, $625$ whose $4$th root equals the number of its divisors:
::$\map {\sigma_0} {6561} = 9 = \sqrt [4] {6561}$
:The $3$rd eighth power after $1$, $256$:
::$6561 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$
:The $4$th power of $9$ after $(1)$, $9$, $81$, $729$:
::$6561 = 9^4$
:The $8$th power of $3$ after $(1)$, $3$, $9$, $27$, $81$, $243$, $729$, $2187$:
::$6561 = 3^8$
:The $9$th fourth power after $1$, $16$, $81$, $256$, $625$, $1296$, $2401$, $4096$:
::$6561 = 9 \times 9 \times 9 \times 9$
:The $81$st square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $5041$, $5184$, $5329$, $5476$, $5625$, $5776$, $5929$, $6084$, $6241$, $6400$
::$6561 = 81 \times 81$
== Also see ==
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Category:Specific Numbers
Category:6561
6569

== Number ==
$6569$ (six thousand, five hundred and sixty-nine) is:
:The $849$th prime number
:The $16$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$, $337$, $991$, $1873$, $2053$, $2377$, $4093$, $4297$, $4583$, $6569$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6569
6578

== Number ==
$6578$ (six thousand, five hundred and seventy-eight) is:
:$2 \times 11 \times 13 \times 23$
:The smallest positive integer which can be expressed as the sum of $3$ fourth powers in $2$ different ways:
::$6578 = 1^4 + 2^4 + 9^4 = 3^4 + 7^4 + 8^4$
== Also see ==
* Positive Integer Sum of 3 Fourth Powers in 2 Ways
== Sources ==
* : $6578$
* : $6578$
Category:Specific Numbers
Category:6578
658

== Number ==
$658$ (six hundred and fifty-eight) is:
:$2 \times 7 \times 47$
:The $53$rd untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $530$, $540$, $552$, $556$, $562$, $576$, $584$, $612$, $624$, $626$, $628$
== Also see ==
* 
Category:Specific Numbers
Category:658
659

== Number ==
$659$ (six hundred and fifty-nine) is:
:The $120$th prime number
:The smaller of the $30$th pair of twin primes:
::$661 - 659 = 2$
:The $30$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $233$, $239$, $251$, $281$, $293$, $359$, $419$, $431$, $443$, $491$, $509$, $593$, $641$, $653$:
::$2 \times 659 + 1 = 1319$, which is prime.
:The $44$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $491$, $499$, $503$, $509$, $541$, $571$, $577$, $593$, $619$, $647$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:659
66

== Number ==
$66$ (sixty-six) is:
:$2 \times 3 \times 11$
:The $2$nd after $55$ of the $3$ repdigit numbers which are also triangular
:The $3$rd sphenic number after $30$, $42$:
::$66 = 2 \times 3 \times 11$
:The $3$rd hexamorphic number after $1$, $45$:
::$66 = H_6$
:The $4$th number after $1$, $3$, $22$ whose divisor sum is square:
:
:The $5$th palindromic triangular number after $0$, $1$, $3$, $6$ whose index is itself palindromic:
::$66 = T_{11}$
:The $6$th hexagonal number after $1$, $6$, $15$, $28$, $45$:
::$66 = 1 + 5 + 9 + 13 + 17 + 21 = 6 \paren {2 \times 6 - 1}$
:The $6$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$
::$66 = \dfrac {11 \times \paren {11 + 1} } 2$
:The $2$nd of those after $55$ which can be split into two palindromic halves
:The $11$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$:
::$66 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = \dfrac {11 \times \paren {11 + 1} } 2$
:The $15$th semiperfect number after $6$, $12$, $18$, $20$, $24$, $28$, $30$, $36$, $40$, $42$, $48$, $54$, $56$, $60$:
::$66 = 11 + 22 + 33$
:The $25$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $27$, $29$, $30$, $36$, $38$, $40$, $43$, $48$, $51$, $53$, $55$, $61$, $62$, $64$, $66$, $\ldots$
:The $38$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $50$, $54$, $55$, $59$, $60$, $61$, $65$ which cannot be expressed as the sum of distinct pentagonal numbers
:The $40$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $37$, $38$, $42$, $43$, $44$, $48$, $49$, $50$, $54$, $55$, $60$, $61$, $65$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* 
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== Sources ==
* : $66$
* : $66$
Category:Specific Numbers
Category:66
66,000,049

== Number ==
$66 \, 000 \, 049$ (sixty-six million and forty-nine) is:
:The $3 \, 896 \, 129$th prime number
:The $25$th minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$, $6469$, $6949$, $9001$, $9049$, $9649$, $9949$, $60 \, 649$, $666 \, 649$, $946 \, 669$, $60 \, 000 \, 049$
== Also see ==
* 
* 
Category:Specific Numbers
Category:66,000,049
66,056,806,460,865,066

== Number ==
$66 \, 056 \, 806 \, 460 \, 865 \, 066$ is:
:$2 \times 3 \times 7 \times 11 \times 1367 \times 6043 \times 17 \, 308 \, 303$
:The $13$th palindromic triangular number after $0$, $1$, $3$, $6$, $66$, $3003$, $66 \, 066$, $617 \, 716$, $3 \, 544 \, 453$, $6 \, 172 \, 882 \, 716$, $30 \, 416 \, 261 \, 403$, $61 \, 728 \, 399 \, 382 \, 716$ whose index is itself palindromic:
::$66 \, 056 \, 806 \, 460 \, 865 \, 066 = T_{363 \, 474 \, 363}$
:The $363 \, 474 \, 363$rd triangular number:
::$66 \, 056 \, 806 \, 460 \, 865 \, 066 = \ds \sum_{k \mathop = 1}^{363 \, 474 \, 363} k = \dfrac {363 \, 474 \, 363 \times \paren {363 \, 474 \, 363 + 1} } 2$
== Also see ==
* 
* 
== Sources ==
* : $363,474,363$
Category:Specific Numbers
Category:66,056,806,460,865,066
66,066

== Number ==
$66 \, 066$ (sixty-six thousand and sixty-six) is:
:$2 \times 3 \times 7 \times 11^2 \times 13$
:The $7$th palindromic triangular number after $0$, $1$, $3$, $6$, $66$, $3003$ whose index is itself palindromic:
::$66 \, 066 = T_{363}$
:The $14$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$, $666$, $3003$, $5995$, $8778$, $15 \, 051$
:The $363$rd triangular number:
::$66 \, 066 = \ds \sum_{k \mathop = 1}^{363} k = \dfrac {363 \times \paren {363 + 1} } 2$
== Also see ==
* 
* 
* 
== Sources ==
* : $363,474,363$
Category:Specific Numbers
Category:66,066
66,600,049

== Number ==
$66 \, 600 \, 049$ (sixty-six million, six hundred thousand and forty-nine) is:
:The $3 \, 929 \, 586$th prime number
:The $26$th (and last) minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$, $6469$, $6949$, $9001$, $9049$, $9649$, $9949$, $60 \, 649$, $666 \, 649$, $946 \, 669$, $60 \, 000 \, 049$, $66 \, 000 \, 049$
== Also see ==
* 
* 
Category:Specific Numbers
Category:66,600,049
660

== Number ==
$660$ (six hundred and sixty) is:
:$2^2 \times 3 \times 5 \times 11$
:The $40$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $210$, $216$, $240$, $288$, $300$, $336$, $360$, $420$, $480$, $504$, $600$, $630$:
::$\map {\sigma_1} {660} = 2016$
== Also see ==
* 
Category:Specific Numbers
Category:660
6601

== Number ==
$6601$ (six thousand, six hundred and one) is:
:$7 \times 23 \times 41$
:The $6$th Carmichael number after $561$, $1105$, $1729$, $2465$, $2821$:
::$\forall a \in \Z: a \perp 6601: a^{6600} \equiv 1 \pmod {6601}$
:The $18$th Poulet number after $341$, $561$, $645$, $1105$, $1387$, $1729$, $1905$, $2047$, $2465$, $2701$, $2821$, $3277$, $4033$, $4369$, $4371$, $4681$, $5461$:
::$2^{6601} \equiv 2 \pmod {6601}$: $6601 = 7 \times 23 \times 41$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6601
661

== Number ==
$661$ (six hundred and sixty-one) is:
:The $121$st prime number
:The lower end of the $8$th record-breaking gap between twin primes:
::$809 - 661 = 148$
:The $15$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$, $160$ such that $m = \ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}! = n! - \paren {n - 1}! + \paren {n - 2}! - \paren {n - 3}! + \cdots \pm 1$ is prime
:The $28$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $353$, $373$, $383$, $433$, $443$, $449$, $499$, $557$, $577$, $599$
:The larger of the $30$th pair of twin primes:
::$661 - 659 = 2$
== Also see ==
* 
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== Sources ==
* : $661$
Category:Specific Numbers
Category:661
6611

== Number ==
$6611$ (six thousand, six hundred and eleven) is:
:$11 \times 601$
:The index of the $5$th Cullen prime after $1$, $141$, $4713$, $5795$:
::$6611 \times 2^{6611} + 1$
== Also see ==
* 
Category:Specific Numbers
Category:6611
6615

== Number ==
$6615$ (six thousand, six hundred and fifteen) is:
:$3^3 \times 5 \times 7^2$
:The $12$th odd abundant number after $945$, $1575$, $2205$, $2835$, $3465$, $4095$, $4725$, $5355$, $5775$, $5985$, $6435$:
::$\map {\sigma_1} {6615} - 6615 = 7065 > 6615$
== Also see ==
* 
Category:Specific Numbers
Category:6615
663

== Number ==
$663$ (six hundred and sixty-three) is:
:$2 \times 13 \times 17$
:The $33$rd Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$, $654$:
::$6 + 6 + 3 = 2 + 1 + 3 + 1 + 7 = 15$
== Also see ==
* 
Category:Specific Numbers
Category:663
664,579

== Number ==
$664 \, 579$ (six hundred and sixty-four thousand, five hundred and seventy-nine) is:
:The $53 \, 911$th prime number
:The number of primes with no more than $7$ digits:
::$2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $\ldots$, $9 \, 999 \, 943$, $9 \, 999 \, 971$, $9 \, 999 \, 973$, $9 \, 999 \, 991$
== Also see ==
* 
* 
== Sources ==
* : $\text I$: The Series of Primes: $1.4$ The sequence of primes
Category:Specific Numbers
Category:664,579
666

== Number ==
$666$ (six hundred and sixty-six) is:
:$2 \times 3^2 \times 37$
:The $3$rd and last after $55$, $66$ of the $3$ repdigit numbers which are also triangular.
:The total of all the entries in a magic square of order $6$, after $1$, $(10)$, $45$, $136$, $325$:
::$666 = \ds \sum_{k \mathop = 1}^{6^2} k = \dfrac {6^2 \paren {6^2 + 1} } 2$
:The $7$th after $4$, $13$, $38$, $87$, $208$, $377$ in the sequence formed by adding the squares of the first $n$ primes:
::$666 = \ds \sum_{i \mathop = 1}^7 {p_i}^2 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2$
:The $8$th positive integer after $1$, $24$, $26$, $87$, $168$, $388$, $594$ whose Euler $\phi$ value is equal to the product of its digits: 
::$\map \phi {666} = 216 = 6 \times 6 \times 6$
:The $9$th palindromic triangular number after $0$, $1$, $3$, $6$, $55$, $66$, $171$, $595$
:The $34$th Smith number after $4$, $22$, $27$, $58$, $\ldots$, $576$, $588$, $627$, $634$, $636$, $645$, $648$, $654$, $663$:
::$6 + 6 + 6 = 2 + 3 + 3 + 3 + 7 = 18$
:The $36$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$:
::$666 = \ds \sum_{k \mathop = 1}^{36} k = \dfrac {36 \times \paren {36 + 1} } 2$
:The basis of an approximation to the Golden Ratio correct to $10$ decimal places:
::$\phi \approx -2 \map \sin {666} = 1.61803 \, 39887 \, 5 \ldots$
== Also see ==
* 
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== Historical Note ==
== Sources ==
* : $\text I$: The Series of Primes: $1.2$ Prime numbers
* : $666$
* : $666$
Category:Specific Numbers
Category:666
666,649

== Number ==
$666 \, 649$ (six hundred and sixty-six thousand, six hundred and forty-nine) is:
:The $54 \, 999$th prime number
:The $22$nd minimal prime base $10$ after $2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$, $6469$, $6949$, $9001$, $9049$, $9649$, $9949$, $60 \, 649$
== Also see ==
* 
* 
Category:Specific Numbers
Category:666,649
666,666

== Number ==
$666 \, 666$ is:
:$2 \times 3^2 \times 7 \times 11 \times 13 \times 37$
:The area of the Pythagorean triangle $693-1924-2045$, the second after the Pythagorean triangle $3-4-5$ to have a repdigit area.
== Also see ==
* 
== Sources ==
* : $666,666$
* : $666,666$
Category:Specific Numbers
Category:666,666
666/Historical Note

== Historical Note on $666$ (Six Hundred and Sixty-Six) ==
$666$ is the famous Number of the Beast of the Book of the Revelation:
:Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.
::::-- The Book of Revelation, chapter 13, verse 18
Hence its reputation as a number traditionally associated with the occult.
Less well known is the fact that the number also appears in the Book of Kings:
:Now the weight of gold that came to Solomon in one year was six hundred threescore and six talents of gold.
::::-- The First Book of Kings, chapter 10, verse 14
It is also the number that can be formed by all the Roman numerals less than $\mathrm M$ used once each:
:$666 = \mathrm {DCLXVI}$
It is possible that $666$ was merely chosen as a representative example of some large unspecified number, in the same way that $101$ or $1001$ is frequently so used in contemporary milieux.
However, its other properties make it far more interesting than that.
It used to be a favourite pastime of Catholic scholars to try to devise alphabet schemes whereby the name was associated with the number $666$, hence supporting their assertion that he was the .
, in return, did the same thing with .
== Sources ==
* : Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory
* : $666$
* : $666$
Category:Historical Notes
6667

== Number ==
$6667$ (six thousand, six hundred and sixty-seven) is:
:$59 \times 113$
:The $10$th tri-automorphic number after $2$, $5$, $7$, $67$, $75$, $92$, $667$, $792$, $875$:
::$6667 \times 3 = 133 \, 34 \mathbf{6 \, 667}$
== Also see ==
* 
== Sources ==
* : $6667$
* : $6667$
Category:Specific Numbers
Category:6667
6667/Mistake

== Source Work ==
:
:The Dictionary
::$6667$
:
:The Dictionary
::$6667$
== Mistake ==
Not a mistake as such, but:
:The patterns appearing in $6667^2$, and similarly in $3334^2$ and so on, are examples of a general rule. Any number, of however many digits, will form a pattern when a sufficiently large number of either $3$s, $6$s or $9$s are prefixed to it. Thus, $72^2 = 5184$, $672^2 = 451, 584$ and $6672^2 = 44, 515, 584$ and so on.
is so vaguely worded as to be all but useless.
== Sources ==
* : $6667$
* : $6667$
Category:Mistakes/Curious and Interesting Numbers
667

== Number ==
$667$ (six hundred and sixty-seven) is:
:$23 \times 29$
:The $7$th tri-automorphic number after $2$, $5$, $7$, $67$, $75$, $92$:
::$667^2 \times 3 = 1 \, 334 \, \mathbf {667}$
== Also see ==
* 
Category:Specific Numbers
Category:667
6670

== Number ==
$6670$ (six thousand, six hundred and seventy) is:
:$2 \times 5 \times 23 \times 29$
:The $58$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $4005$, $4186$, $4371$, $4560$, $4950$, $5151$, $5356$, $5565$, $5778$, $5995$, $6216$, $6441$:
::$6670 = \ds \sum_{k \mathop = 1}^{58} \paren {4 k - 3} = 58 \paren {2 \times 58 - 1}$
:The $115$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $5995$, $6105$, $6216$, $6328$, $6441$, $6555$:
::$6670 = \ds \sum_{k \mathop = 1}^{115} k = \dfrac {115 \times \paren {115 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:6670
6682

== Number ==
$6682$ (six thousand, six hundred and eighty-two) is:
:$2 \times 13 \times 257$
:The $52$nd heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $5452$, $5688$, $5929$, $6175$, $6426$:
::$6682 = \ds \sum_{k \mathop = 1}^{52} \paren {5 k - 4} = \dfrac {52 \paren {5 \times 52 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:6682
67

== Number ==
$67$ (sixty-seven) is:
:The $19$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$, $61$
:The $4$th tri-automorphic number after $2$, $5$, $7$:
::$67^2 \times 3 = 13 \, 4 \mathbf {67}$
:The $6$th prime number after $2$, $3$, $5$, $7$, $23$ consisting of a string of consecutive ascending digits
:The $10$th prime $p$ after $11$, $23$, $29$, $37$, $41$, $43$, $47$, $53$, $59$ such that the Mersenne number $2^p - 1$ is composite
:The index of the $10$th Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$ which asserted to be prime
::(in this case he was not correct: he missed $61$, and $2^{67} - 1 = 193 \, 707 \, 721 \times 761 \, 838 \, 257 \, 287$)
:The $12$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$
:The $13$th positive integer $n$ after $5$, $11$, $17$, $23$, $29$, $30$, $36$, $42$, $48$, $54$, $60$, $61$ such that no factorial of an integer can end with $n$ zeroes
:The $16$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $\ldots$
:The $24$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $67$, $\ldots$
:The $29$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $\ldots$, $35$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $\ldots$
:The $31$st integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{67} = 147 \, 573 \, 952 \, 589 \, 676 \, 412 \, 928$
:The $39$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $50$, $54$, $55$, $59$, $60$, $61$, $65$, $66$ which cannot be expressed as the sum of distinct pentagonal numbers
:The $41$st (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $55$, $60$, $61$, $65$, $66$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* Factors of Mersenne Number M67
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Category:Specific Numbers
Category:67
67,280,421,310,721

== Number ==
$67,280,421,310,721$ is:
:The $n$th prime, where $n$ remains to be determined
:One of the two prime factors of the $6$th Fermat number $2^{\paren {2^6} } + 1$, the other being $274 \, 177$:
::$67 \, 280 \, 421 \, 310 \, 721 = 262 \, 814 \, 145 \, 745 \times 2^8 + 1 = 5 \times 47 \times 373 \times 2998279 \times 2^8 + 1$
:The largest prime to have been identified as such without the use of a computer. This was done by subsequent to his factorisation of $F_6$ in $1880$.
== Also see ==
* 
* 
== Sources ==
* : $257$
* : $257$
Category:Specific Numbers
Category:67,280,421,310,721
671

== Number ==
$671$ (six hundred and seventy-one) is:
:$11 \times 61$
:The $3$rd integer after $3$, $228$ whose square is the sum of $2$ coprime cubes:
::$671^2 = 65^3 + 56^3$
:The $4$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$:
::$3^{671} \equiv 3 \pmod {671}$
:The magic constant of a magic square of order $11$, after $1$, $(5)$, $15$, $34$, $65$, $111$, $175$, $260$, $369$, $505$:
::$671 = \ds \dfrac 1 {11} \sum_{k \mathop = 1}^{11^2} k = \dfrac {11 \paren {11^2 + 1} } 2$
== Also see ==
* 
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Category:Specific Numbers
Category:671
672

== Number ==
$672$ (six hundred and seventy-two) is:
:$2^5 \times 3 \times 7$
:The $2$nd triperfect number after $120$:
::$\map {\sigma_1} {672} = 2016 = 3 \times 672$
:The $7$th Ore number after $1$, $6$, $28$, $140$, $270$, $496$:
::$\dfrac {672 \times \map {\sigma_0} {672} } {\map {\sigma_1} {672} } = 8$
:and the $5$th after $1$, $6$, $14$, $270$ whose divisors also have an arithmetic mean which is an integer:
::$\dfrac {\map {\sigma_1} {672} } {\map {\sigma_0} {672} } = 84$
:The $14$th positive integer after $128$, $192$, $256$, $288$, $320$, $384$, $432$, $448$, $480$, $512$, $576$, $640$, $648$ with $7$ or more prime factors:
::$672 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 7$
:The $21$st second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $222$, $260$, $301$, $345$, $392$, $442$, $495$, $551$, $610$:
::$672 = \dfrac {21 \paren {3 \times 21 + 1} } 2$
:The $32$nd Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $216$, $224$, $312$, $315$, $384$, $432$, $612$, $624$:
::$672 = 8 \times 84 = 8 \times \paren {6 \times 7 \times 2}$
:The $42$nd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $392$, $425$, $442$, $477$, $495$, $532$, $551$, $590$, $610$, $651$:
::$672 = \dfrac {21 \paren {3 \times 21 + 1} } 2$
== Also see ==
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== Sources ==
* : $672$
* : $672$
Category:Specific Numbers
Category:672
6724

== Number ==
$6724$ (six thousand, seven hundred and twenty-four) is:
:$2^2 \times 41^2$
:The $82$nd square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $5041$, $5184$, $5329$, $5476$, $5625$, $5776$, $5929$, $6084$, $6241$, $6400$, $6561$
::$6724 = 82 \times 82$
== Also see ==
* 
Category:Specific Numbers
Category:6724
6729

== Number ==
$6729$ (six thousand, seven hundred and twenty-nine) is:
:$3 \times 2243$
:$6729$ and its double contain all the digits from $1$ to $9$ between them:
::$2 \times 6729 = 13 \, 458$
== Also see ==
* Integer and its Double forming Pandigital Pair
== Sources ==
* : $6729$
* : $6729$
Category:Specific Numbers
Category:6729
673

== Number ==
$673$ (six hundred and seventy-three) is:
:The $122$nd prime number
:The $15$th Proth prime after $3$, $5$, $13$, $17$, $41$, $97$, $113$, $193$, $241$, $257$, $353$, $449$, $577$, $641$:
::$673 = 21 \times 2^5 + 1$
:The $41$st left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $443$, $467$, $523$, $547$, $613$, $617$, $643$, $647$, $653$
== Also see ==
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Category:Specific Numbers
Category:673
675

== Number ==
$675$ (six hundred and seventy-five) is:
:$3^3 \times 5^2$
:The smaller of the $3$rd pair of consecutive powerful numbers:
::$675 = 3^3 \times 5^2$, $676 = 2^2 \times 13^2$
:The only known such pair whose first element is odd.
:The $44$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $400$, $432$, $441$, $484$, $500$, $512$, $529$, $576$, $625$, $648$:
::$675 = 3^3 \times 5^2$
== Also see ==
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== Sources ==
* : $675$
Category:Specific Numbers
Category:675
676

== Number ==
$676$ (six hundred and seventy-six) is:
:$2^2 \times 13^2$
:The $1$st palindromic square with a non-palindromic square root:
::$676 = 26^2$
:The larger of the $3$rd pair of consecutive powerful numbers:
::$675 = 3^3 \times 5^2$, $676 = 2^2 \times 13^2$
:The only known such pair whose second element is even.
:The $15$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $121$, $144$, $225$, $441$, $484$ which has no more than $2$ distinct digits and does not end in $0$:
::$676 = 26^2$
:The $22$nd positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $130$, $169$, $196$, $214$, $226$, $289$, $324$, $370$, $400$, $526$, $529$, $625$, $676$, $\ldots$
:The $26$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $361$, $400$, $441$, $484$, $529$, $576$, $625$:
::$676 = 26 \times 26$
:The $45$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $400$, $432$, $441$, $484$, $500$, $512$, $529$, $576$, $625$, $648$, $675$:
::$676 = 2^2 \times 13^2$
== Also see ==
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== Sources ==
* : $676$
* : $676$
Category:Specific Numbers
Category:676
6765

== Number ==
$6765$ (six thousand, seven hundred and sixty-five) is:
:$3 \times 5 \times 11 \times 41$
:The $20$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$, $55$, $89$, $144$, $233$, $377$, $610$, $987$, $1597$, $2584$, $4181$:
::$6765 = 2584 + 4181$
== Also see ==
* 
Category:Specific Numbers
Category:6765
677

== Number ==
$677$ (six hundred and seventy-seven) is:
:The $123$rd prime number
:The $10$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$, $401$, $577$:
::$677 = 26^2 + 1$
:The $19$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 18^2 + 29 = 677$ ()
:The $29$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $353$, $373$, $383$, $433$, $443$, $449$, $499$, $557$, $577$, $599$, $661$
== Also see ==
* 
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Category:Specific Numbers
Category:677