nyuuzyou/wikis huggingface dataset:
16

== Number ==
$16$ (sixteen) is:
:$2^4$
:The $1$st power of $16$ after the zeroth $1$:
::$16 = 16^1$
:The $1$st square number to be the sum of two triangular numbers in two different ways:
::$16 = 10 + 6 = 15 + 1$
:The $2$nd fourth power after $1$:
::$16 = 2 \times 2 \times 2 \times 2$
:The $2$nd power of $4$ after $(1)$, $4$:
::$16 = 4^2$
:The number of different binary operations that can be applied to a set with $2$ elements
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $3$ different ways:
::$16 = 1 + 15 = 3 + 13 = 7 + 9$
:The $4$th square number after $1$, $4$, $9$:
::$16 = 4 \times 4$
:The $4$th power of $2$ after $(1)$, $2$, $4$, $8$:
::$16 = 2^4$
:The $4$th square after $1$, $4$, $9$ which has no more than $2$ distinct digits and does not end in $0$:
::$16 = 4^2$
:The $5$th almost perfect number after $1$, $2$, $4$, $8$:
::$\map {\sigma_1} {16} = 31 = 2 \times 16 - 1$
:The $5$th powerful number after $1$, $4$, $8$, $9$
:The $6$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
::$3$, $4$, $9$, $10$, $12$, $16$, $\ldots$
:The $8$th even number after $2$, $4$, $6$, $8$, $10$, $12$, $14$ which cannot be expressed as the sum of $2$ composite odd numbers.
:The $9$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$:
::$\map {\sigma_1} {16} = 31$
:The $9$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$:
::$16 = 3 + 13$
:The $11$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.
:The $11$th positive integer after $2$, $3$, $4$, $7$, $8$, $9$, $10$, $11$, $14$, $15$ which cannot be expressed as the sum of distinct pentagonal numbers.
:The $12$th after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $10$, $12$, $14$ of $21$ integers which can be represented as the sum of two primes in the maximum number of ways
:The $14$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$ such that $2^n$ contains no zero in its decimal representation:
::$2^{16} = 65 \, 536$
== Also see ==
* Definition:Hexadecimal Notation
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $\S 13$: The fundamental theorem of arithmetic
* : $16$
* : $16$
Category:Specific Numbers
Category:16
16,057

== Number ==
$16 \, 057$ (sixteen thousand and fifty-seven) is:
:The $1686$th prime number
:The $4$th after $587$, $1777$, $5347$ of a sequence of $11$ primes formed by tripling and adding $16$:
::$16 \, 057 = 3 \times 5347 + 16$
== Also see ==
* 
* 
Category:Specific Numbers
Category:16,057
16,384

== Number ==
$16 \, 384$ (sixteen thousand, three hundred and eighty-four) is:
:$2^{14}$
:The $4$th seventh power after $1$, $128$, $2187$:
::$16 \, 384 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$
:The $7$th power of $4$ after $(1)$, $4$, $16$, $64$, $256$, $1024$, $4096$:
::$16 \, 384 = 4^7$
:The $14$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$, $8192$:
::$16 \, 384 = 2^{14}$
:The $15$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$, $128$, $256$, $512$, $1024$, $2048$, $4096$, $8192$:
::$\map {\sigma_1} {16 \, 384} = 32 \, 767 = 2 \times 16 \, 384 - 1$
:The $128$th square number:
::$16 \, 384 = 128 \times 128$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:16,384
16,470

== Number ==
$16 \, 470$ (sixteen thousand, four hundred and seventy) is:
:$2 \times 3^3 \times 5 \times 61$
:The $2$nd self-locating number in $\pi$ after $1$
== Also see ==
* 
== Sources ==
* : Cool Numbers: Self-location and $\pi$
Category:16,470
Category:Specific Numbers
16,777,216

== Number ==
$16 \, 777 \, 216$ (sixteen million, seven hundred and seventy-seven thousand, two hundred and sixteen) is:
:$2^{24}$
:The $6$th power of $16$ after $(1)$, $16$, $256$, $4096$, $65 \, 536$, $1 \, 048 \, 576$:
::$16 \, 777 \, 216 = 16^6$
:The $8$th eighth power after $1$, $256$, $6561$, $65 \, 536$, $390 \, 625$, $1 \, 679 \, 616$, $5 \, 764 \, 801$:
::$16 \, 777 \, 216 = 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8 \times 8$
:The $8$th power of $8$ after $(1)$, $8$, $64$, $512$, $4096$, $32 \, 768$, $262 \, 144$, $2 \, 097 \, 152$:
::$16 \, 777 \, 216 = 8^8$
:The $12$th power of $4$ after $(1)$, $4$, $16$, $64$, $256$, $1024$, $4096$, $16 \, 384$, $65 \, 536$, $262 \, 144$, $1 \, 048 \, 576$, $4 \, 194 \, 304$:
::$16 \, 777 \, 216 = 4^{12}$
:The $16$th sixth power after $1$, $64$, $729$, $4096$, $15 \, 625$, $\ldots$, $2 \, 985 \, 984$, $4 \, 826 \, 809$, $7 \, 529 \, 536$, $11 \, 390 \, 625$:
::$16 \, 777 \, 216 = 16 \times 16 \times 16 \times 16 \times 16 \times 16$
:The $24$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $\ldots$, $131 \, 072$, $262 \, 144$, $524 \, 288$, $1 \, 048 \, 576$, $2 \, 097 \, 152$, $4 \, 194 \, 304$, $8 \, 388 \, 608$:
::$16 \, 777 \, 216 = 2^{24}$
:The $25$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $\ldots$, $131 \, 072$, $262 \, 144$, $524 \, 288$, $1 \, 048 \, 576$, $2 \, 097 \, 152$, $4 \, 194 \, 304$, $8 \, 388 \, 608$:
::$\map {\sigma_1} {16 \, 777 \, 216} = 33 \, 554 \, 431 = 2 \times 16 \, 777 \, 216 - 1$
:The $64$th fourth power:
::$16 \, 777 \, 216 = 64 \times 64 \times 64 \times 64$
:The $256$th cube number:
::$16 \, 777 \, 216 = 256 \times 256 \times 256$
:The $4096$th square number:
::$16 \, 777 \, 216 = 4096 \times 4096$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:16,777,216
16,796

== Number ==
$16 \, 796$ (sixteen thousand, seven hundred and ninety-six) is:
:$2^2 \times 13 \times 17 \times 19$
:The $10$th Catalan number after $(1)$, $1$, $2$, $5$, $14$, $42$, $132$, $429$, $1430$, $4862$:
::$16 \, 796 = \dfrac 1 {10 + 1} \dbinom {2 \times 10} {10} = \dfrac 1 {11} \times 184 \, 756$
== Also see ==
* 
Category:Specific Numbers
Category:16,796
16,807

== Number ==
$16 \, 807$ (sixteen thousand, eight hundred and seven) is:
:$7^5$
:The $5$th power of $7$ after $(1)$, $7$, $49$, $343$, $2401$:
::$16 \, 807 = 7^5$
:The $7$th fifth power after $1$, $32$, $243$, $1024$, $3125$, $7776$:
::$16 \, 807 = 7 \times 7 \times 7 \times 7 \times 7$
== Also see ==
* 
* 
Category:Specific Numbers
Category:16,807
16,830

== Number ==
$16 \, 830$ (sixteen thousand, eight hundred and thirty) is:
:$2 \times 3^2 \times 5 \times 11 \times 17$
:$16 \, 830^3 = \ds \sum_{k \mathop = 1134}^{2133} k^3$
== Also see ==
* Cube Numbers as Sum of Sequence of Cubes
== Sources ==
* : $16,830$
* : $16,830$
Category:Specific Numbers
Category:16,830
16,843

== Number ==
$16 \, 843$ (sixteen thousand, eight hundred and forty-three) is:
:The $1944$th prime number
:The smaller of the two known Wolstenholme primes, the other being $2 \, 124 \, 679$
== Also see ==
* Charles Babbage's Conjecture
* 
* 
== Sources ==
* : $16,843$
* : $16,843$
Category:Specific Numbers
Category:16,843
160

== Number ==
$160$ (one hundred and sixty) is:
:$2^5 \times 5$
:The $3$rd of the $4$ cubic recurring digital invariants after $55, 136$:
::$160 \to 217 \to 352 \to 160$
:The $5$th positive integer after $64$, $96$, $128$, $144$ with $6$ or more prime factors:
::$160 = 2 \times 2 \times 2 \times 2 \times 2 \times 5$
:The $14$th integer $n$ after $3$, $4$, $5$, $6$, $7$, $8$, $10$, $15$, $19$, $41$, $59$, $61$, $105$ 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
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:160
160,426,514

== Number ==
$160 \, 426 \, 514$ is:
:$2 \times 1361 \times 58 \, 937$
:The smallest positive integer which can be expressed as the sum of $3$ sixth powers in $2$ different ways:
:and also:
== Also see ==
* Smallest Integer which is Sum of 3 Sixth Powers in 2 Ways
== Sources ==
* : $160,426,514$
* : $160,426,514$
Category:Specific Numbers
Category:160,426,514
1600

== Number ==
$1600$ (one thousand six hundred) is:
:$2^6 \times 5^2$
:The $15$th square number after $1$, $4$, $36$, $121$, $144$, $256$, $324$, $400$, $576$, $784$, $900$, $961$, $1024$, $1296$ to be the divisor sum value of some (strictly) positive integer:
::$1600 = \map {\sigma_1} {1029} = \map {\sigma_1} {1393} = \map {\sigma_1} {1501}$
:The $40$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1024$, $1089$, $1156$, $1225$, $1296$, $1369$, $1444$, $1521$:
::$1600 = 40 \times 40$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1600
1601

== Number ==
$1601$ (one thousand, six hundred and one) is:
:The $252$nd prime number
:The $12$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$, $37$, $101$, $197$, $257$, $401$, $577$, $677$, $1297$:
::$1601 = 40^2 + 1$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1601
161

== Number ==
$161$ (one hundred and sixty-one) is:
:$7 \times 23$
:The $6$th hexagonal pyramidal number after $1$, $7$, $22$, $50$, $95$:
::$161 = \ds \sum_{k \mathop = 1}^6 k \paren {2 k - 1} = \dfrac {6 \paren {6 + 1} \paren {4 \times 6 - 1} } 6$
:The $6$th Cullen number after $1$, $3$, $9$, $25$, $65$:
::$161 = 5 \times 2^5 + 1$
:The $7$th inconsummate number after $62$, $63$, $65$, $75$, $84$, $95$:
::$\nexists n \in \Z_{>0}: n = 161 \times \map {s_{10} } n$
:The $32$nd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes
:The $60$th and largest (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $95$, $96$, $102$, $108$, $114$, $119$, $120$, $125$, $143$, $155$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* 
* 
* 
* 
* 
== Sources ==
* : $161$
* : $161$
Category:Specific Numbers
Category:161
161,038

== Number ==
$161 \, 038$ (one hundred and sixty-one thousand and thirty-eight) is:
:$2 \times 73 \times 1103$
:The $1$st even Poulet number:
::$2^{161 \, 038} \equiv 2 \pmod {161 \, 038}$: $161 \, 038 = 2 \times 73 \times 1103$
== Also see ==
* 
== Sources ==
* : $161,038$
* : $161,038$
Category:Specific Numbers
Category:161,038
161,051

== Number ==
$161 \, 051$ (one hundred and sixty-one thousand and fifty-one) is:
:$11^5$
:The $11$th fifth power after $1$, $32$, $243$, $1024$, $3125$, $7776$, $16 \, 807$, $32 \, 768$, $59 \, 049$, $100 \, 000$:
::$161 \, 051 = 11 \times 11 \times 11 \times 11 \times 11$
:The $5$th power of $11$ after $(1)$, $11$, $121$, $1331$, $14 \, 641$:
::$161 \, 051 = 11^5$
== Also see ==
* 
* 
Category:Specific Numbers
Category:161,051
1610/Historical Note

== Historical Note on 1610 ==
$1610$ was the year when the the King James (Authorized) Version of the Christian Bible was completed.
Consider the $46$th psalm, as rendered in this King James version.
The $46$th word of this psalm is shake.
The $46$th word from the end, counting backwards, is spear.
This was when William Shakespeare was $46$ years old.
$1610 = 35 \times 46$.
== Sources ==
* : $46$
* : $46$
Category:Historical Notes
1613

== Number ==
$1613$ (one thousand six hundred and thirteen) is:
:The $255$th prime number
:The $60$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $983$, $997$, $1223$, $1283$, $1367$, $1373$, $1523$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1613
1617

== Number ==
$1617$ (one thousand, six hundred and seventeen) is:
:$3 \times 7^2 \times 11$
:The $33$rd pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $925$, $1001$, $1080$, $1162$, $1247$, $1335$, $1426$, $1520$:
::$1617 = \ds \sum_{k \mathop = 1}^{33} \paren {3 k - 2} = \dfrac {33 \paren {3 \times 33 - 1} } 2$
:The $65$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1335$, $1365$, $1426$, $1457$, $1520$, $1552$:
::$1617 = \ds \sum_{k \mathop = 1}^{33} \paren {3 k - 2} = \dfrac {33 \paren {3 \times 33 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1617
1619

== Number ==
$1619$ (one thousand six hundred and nineteen) is:
:The $256$th prime number
:The $35$th safe prime after $5$, $7$, $11$, $23$, $47$, $\ldots$, $839$, $863$, $887$, $983$, $1019$, $1187$, $1283$, $1307$, $1319$, $1367$, $1439$, $1487$, $1523$:
::$1523 = 2 \times 761 + 1$
:The smaller of the $52$nd pair of twin primes:
::$1621 - 1619 = 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1619
162, 259,276, 829,213, 363,391, 578,010, 288,127

== Number ==
$162 \, 259 \, 276 \, 829 \, 213 \, 363 \, 391 \, 578 \, 010 \, 288 \, 127$ is:
:A prime number
:The $11$th Mersenne prime:
::$162 \, 259 \, 276 \, 829 \, 213 \, 363 \, 391 \, 578 \, 010 \, 288 \, 127 = 2^{107} - 1$
== Historical Note ==
== Also see ==
* 
* 
Category:Specific Numbers
Category:162, 259,276, 829,213, 363,391, 578,010, 288,127
162, 259,276, 829,213, 363,391, 578,010, 288,127/Historical Note

== Historical Note on $162 \, 259 \, 276 \, 829 \, 213 \, 363 \, 391 \, 578 \, 010 \, 288 \, 127$ ==
 announced in $1914$ that he had discovered that $162 \, 259 \, 276 \, 829 \, 213 \, 363 \, 391 \, 578 \, 010 \, 288 \, 127 = 2^{107} - 1$ is prime.
 simultaneously made the same announcement at the same time.
As some of 's claims of the primality of other Mersenne numbers proved to be incorrect, this result is usually attributed to rather than to .
It was the $12$th Mersenne prime to be discovered, although the $11$th in sequence.
 had demonstrated in $1876$ that $2^{127} - 1$ (actually the $12$th in sequence) is prime.
== Sources ==
* 
* : Preface
Category:Historical Notes
163

== Number ==
$163$ (one hundred and sixty-three) is:
:The $38$th prime number
:The $9$th prime number after $53$, $71$, $103$, $107$, $109$, $149$, $151$, $157$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$
:The $33$rd lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $115$, $127$, $129$, $133$, $135$, $141$, $151$, $159$, $163$, $\ldots$
:This expression relating $e$ and $\pi$ comes within $10^{-12}$ of an integer:
::$e^{\pi \sqrt {163} } \approx 262 \, 537 \, 412 \, 640 \, 768 \, 743 \cdotp 99999 \, 99999 \, 99250 \ldots$
== Also see ==
* Approximate Relations between $\pi$ and $e$: $e^{\pi \sqrt{163} }$
* 
* 
* 
== Sources ==
* : $163$
* : $163$
Category:Specific Numbers
Category:163
1634

== Number ==
$1634$ (one thousand, six hundred and thirty-four) is:
:$2 \times 19 \times 43$
:The $14$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $153$, $370$, $371$, $407$:
::$1^4 + 6^4 + 3^4 + 4^4 = 1 + 1296 + 81 + 256 = 1634$
== Also see ==
* 
== Sources ==
* : $1634$
* : $1634$
Category:Specific Numbers
Category:1634
1638

== Number ==
$1638$ (one thousand, six hundred and thirty-eight) is:
:$2 \times 3^2 \times 7 \times 13$
:The $8$th Ore number after $1$, $6$, $28$, $140$, $270$, $496$, $672$:
::$\dfrac {1638 \times \map {\sigma_0} {1638} } {\map {\sigma_1} {1638} } = 9$
:and the $6$th after $1$, $6$, $14$, $270$, $672$ whose divisors also have an arithmetic mean which is an integer:
::$\dfrac {\map {\sigma_1} {1638} } {\map {\sigma_0} {1638} } = 84$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1638
164

== Number ==
$164$ (one hundred and sixty-four) is:
:$2^2 \times 41$
:The $5$th positive integer solution after $1, 3, 15, 104$ to $\map \phi n = \map \phi {n + 1}$:
::$\map \phi {164} = 80 = \map \phi {165}$
== Also see ==
* 
Category:Specific Numbers
Category:164
164,597,832

== Number ==
$164 \, 597 \, 832$ is:
:$2^3 \times 3^4 \times 7 \times 131 \times 277$
:A penholodigital integer which can be expressed as the product of a penholodigital set:
::$164 \, 597 \, 832 = 6 \times 7 \times 9 \times 524 \times 831$
== Also see ==
* Penholodigital Product of Penholodigital Set of Factors
== Sources ==
* : $164,597,832$
Category:Specific Numbers
Category:164,597,832
1648

== Number ==
$1648$ (one thousand six hundred and forty-eight) is:
:$2^4 \times 103$
:With $1648$, an element of the $4$th quasiamicable pair:
::$\map {\sigma_1} {1575} = \map {\sigma_1} {1648} = 3224 = 1575 + 1648 + 1$
== Also see ==
* 
Category:Specific Numbers
Category:1648
165

== Number ==
$165$ (one hundred and sixty-five) is:
:$3 \times 5 \times 11$
:The $13$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$, $130$, $138$, $154$:
::$165 = 3 \times 5 \times 11$
:The $9$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$:
::$165 = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 = \dfrac {9 \paren {9 + 1} \paren {9 + 2} } 6$
:The $1$st of the $2$nd ordered triple of consecutive integers after $\tuple {105, 106, 107}$ that have Euler $\phi$ values which are strictly increasing:
::$\map \phi {165} = 80$, $\map \phi {166} = 82$, $\map \phi {167} = 166$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:165
1650

== Number ==
$1650$ (one thousand, six hundred and fifty) is:
:$2 \times 3 \times 5^2 \times 11$
:The $33$rd second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $610$, $672$, $737$, $805$, $876$, $950$, $1027$, $1107$, $1190$, $1276$, $1365$, $1457$, $1552$:
::$1650 = \dfrac {33 \paren {3 \times 33 + 1} } 2$
:The $66$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $1335$, $1365$, $1426$, $1457$, $1520$, $1552$, $1617$:
::$1650 = \dfrac {33 \paren {3 \times 33 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1650
1651

== Number ==
$1651$ (one thousand, six hundred and fifty-one) is:
:$13 \times 127$
:The $26$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $874$, $970$, $1071$, $1177$, $1288$, $1404$, $1525$:
::$1651 = \ds \sum_{k \mathop = 1}^{26} \paren {5 k - 4} = \dfrac {26 \paren {5 \times 26 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:1651
1653

== Number ==
$1653$ (one thousand, six hundred and fifty-three) is:
:$3 \times 19 \times 29$
:The $29$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $703$, $780$, $861$, $946$, $1035$, $1225$, $1326$, $1431$, $1540$:
::$1653 = \ds \sum_{k \mathop = 1}^{29} \paren {4 k - 3} = 29 \paren {2 \times 29 - 1}$
:The $57$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $990$, $1035$, $1081$, $1128$, $1176$, $1225$, $1275$, $1326$, $1378$, $1431$, $1485$, $1540$, $1596$:
::$1653 = \ds \sum_{k \mathop = 1}^{57} k = \dfrac {57 \times \paren {57 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1653
1657

== Number ==
$1657$ (one thousand, six hundred and fifty-seven) is:
:The $260$th prime number
:The $24$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $817$, $919$, $1027$, $1141$, $1261$, $1387$, $1519$:
::$1657 = \ds 1 + \sum_{k \mathop = 1}^{24 - 1} 6 k = 24^3 - 23^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1657
166

== Number ==
$166$ (one hundred and sixty-six) is:
:$2 \times 83$
:The $2$nd of the $2$nd ordered triple of consecutive integers after $\tuple {105, 106, 107}$ that have Euler $\phi$ values which are strictly increasing:
::$\map \phi {165} = 80$, $\map \phi {166} = 82$, $\map \phi {167} = 166$
:The $8$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$, $121$:
::$1 + 6 + 6 = 2 + 8 + 3 = 13$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $11$ different ways
:The $12$th integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$, $38$, $94$, $166$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:166
1669

== Number ==
$1669$ (one thousand, six hundred and sixty-nine) is:
:The $263$rd prime number
:The $8$th after $199$, $409$, $619$, $829$, $1039$, $1249$, $1459$ of the smallest $10$ primes in arithmetic sequence:
::$1669 = 7 \times 210 + 199$
:The larger of the $53$rd pair of twin primes:
::$1669 - 1667 = 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1669
167

== Number ==
$167$ (one hundred and sixty-seven) is:
:The $39$th prime number
:The $3$rd of the $2$nd ordered triple of consecutive integers after $\tuple {105, 106, 107}$ that have Euler $\phi$ values which are strictly increasing:
::$\map \phi {165} = 80$, $\map \phi {166} = 82$, $\map \phi {167} = 166$
:The $9$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$:
::$167 = 2 \times 83 + 1$
:The $10$th prime number after $53$, $71$, $103$, $107$, $109$, $149$, $151$, $157$, $163$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$
:The $13$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$, $157$
:The $14$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$, $113$, $131$, $149$
:The $18$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$
:The $27$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $91$, $94$, $97$, $100$, $103$, $109$, $129$, $130$, $133$, $139$:
::$167 \to 1^2 + 6^2 + 7^2 = 1 + 36 + 49 = 86 \to 8^2 + 6^2 = 64 + 36 = 100 \to 1^2 + 0^2 + 0^2 = 1$
:The $33$rd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:167
167,772,161

== Number ==
$167 \, 772 \, 161$ is:
:The $10 \, 385 \, 401$st prime.
:One of the prime factors of the $24$th Fermat number $2^{\left({2^{23} }\right)} + 1$:
::$167 \, 772 \, 161 = 5 \times 2^{25} + 1$
== Also see ==
* 
Category:Specific Numbers
Category:167,772,161
1676

== Number ==
$1676$ (one thousand six hundred and seventy-six) is:
:$2^2 \times 419$
:The $17$th number after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $89$, $135$, $175$, $518$, $598$, $1306$ which is the sum of the increasing powers of its digits taken in order:
::$1^1 + 6^2 + 7^3 + 6^4 = 1676$
== Also see ==
* 
Category:Specific Numbers
Category:1676
168

== Number ==
$168$ (one hundred and sixty-eight) is:
:$2^3 \times 3 \times 7$
:The $1$st element of the $1$st set of $3$ integers $T$ such that $m \map {\sigma_0} m$ is equal for each $m \in T$:
::$168 \times \map {\sigma_0} {168} = 192 \times \map {\sigma_0} {192} = 224 \times \map {\sigma_0} {224} = 2688$
:The number of primes with no more than $3$ digits:
::$2, 3, 5, 7, 11, 13, 17, 19, 23, \ldots, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997$
:The $5$th positive integer after $1$, $24$, $26$, $87$ whose Euler $\phi$ value is equal to the product of its digits: 
::$\map \phi {168} = 48 = 1 \times 6 \times 8$
:The $11$th integer $n$ after $1, 3, 15, 30, 35, 56, 70, 78, 105, 140$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {168} = 16$, $\map \phi {168} = 48$, $\map {\sigma_1} {168} = 480$
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $13$ ways.
:The $25$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$, $18$, $20$, $24$, $30$, $36$, $42$, $48$, $60$, $72$, $84$, $90$, $96$, $108$, $120$, $144$:
::$\map {\sigma_1} {168} = 480$
== Also see ==
* 
* 
* 
* 
* 
* 
== Sources ==
* : $168$
Category:Specific Numbers
Category:168
1680

== Number ==
$1680$ (one thousand, six hundred and eighty) is:
:$2^4 \times 3 \times 5 \times 7$
:The $17$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$:
::$\map {\sigma_0} {1680} = 40$
:The $17$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$, $1260$:
::$\dfrac {\map {\sigma_1} {1680} } {1680} = \dfrac {5952} {1680} \approx 3 \cdotp 543$
:The $24$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $645$, $736$, $833$, $936$, $1045$, $1160$, $1281$, $1408$, $1541$:
::$1680 = \ds \sum_{k \mathop = 1}^{24} \paren {6 k - 5} = 24 \paren {3 \times 24 - 2}$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1680