nyuuzyou/wikis huggingface dataset:
122

== Number ==
$122$ (one hundred and twenty-two) is:
:$2 \times 61$
:The $16$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$, $86$, $90$, $94$, $98$, $114$, $118$:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 122$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The $10$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$, $116$:
::$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 122$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The $2$nd even integer that cannot be expressed as the sum of $2$ prime numbers of which the smaller one is $3$, $5$ or $7$.
:The $23$rd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
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Category:Specific Numbers
Category:122
122,410

== Number ==
$122 \, 410$ (one hundred and twenty-two thousand, four hundred and ten) is:
:$2 \times 5 \times 12 \, 241$
:The $21$st term of a sociable chain of order $28$ whose smallest term is $14 \, 316$:
::$14 \, 316$, $19 \, 116$, $31 \, 704$, $47 \, 616$, $83 \, 328$, $177 \, 792$, $295 \, 488$, $629 \, 072$, $589 \, 786$, $294 \, 896$, $358 \, 336$, $418 \, 904$, $366 \, 556$, $274 \, 924$, $275 \, 444$, $243 \, 760$, $376 \, 736$, $381 \, 028$, $285 \, 778$, $152 \, 990$, $122 \, 410$, $97 \, 946$, $48 \, 976$, $45 \, 946$, $22 \, 976$, $22 \, 744$, $19 \, 916$, $17 \, 716$, $14 \, 316$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:122,410
1223

== Number ==
$1223$ (one thousand, two hundred and twenty-three) is:
:The $200$th prime number
:The $19$th balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$, $563$, $593$, $607$, $653$, $733$, $947$, $977$, $1103$, $1123$, $1187$:
::$1223 = \dfrac {1217 + 1229} 2$
:The $43$rd Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $761$, $809$, $911$, $953$, $1013$, $1019$, $1031$, $1049$, $1103$:
::$2 \times 1223 + 1 = 2447$, which is prime.
:The $55$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $743$, $773$, $797$, $823$, $853$, $883$, $937$, $947$, $953$, $967$, $983$, $997$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1223
1225

== Number ==
$1225$ (one thousand, two hundred and twenty-five) is:
:$5^2 \times 7^2$
:The $2$nd square number after $1$ which can be expressed as the sum of a sequence of odd cubes from $1$:
::$1225 = 35^2 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3$
:The $3$rd number after $1$, $36$ to be both square and triangular:
::$1225 = 35^2 = \dfrac {49 \times \paren {49 + 1} } 2$
:The $4$th hexamorphic number after $1$, $45$, $66$:
::$1225 = H_{25}$
:The $4$th square after $49$, $169$, $361$ whose decimal representation can be split into two parts which are each themselves square:
::$1225 = 35^2$; $1 = 1^2$, $225 = 15^2$
:The total of all the entries in a magic square of order $7$, after $1$, $(10)$, $45$, $136$, $325$, $666$:
::$1225 = \ds \sum_{k \mathop = 1}^{7^2} k = \dfrac {7^2 \paren {7^2 + 1} } 2$
:The $25$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $703$, $780$, $861$, $946$, $1035$, $1128$:
::$1225 = \ds \sum_{k \mathop = 1}^{25} \paren {4 k - 3} = 25 \paren {2 \times 25 - 1}$
:The $30$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $706$, $730$, $771$, $784$, $841$, $1024$, $1089$, $1225$, $\ldots$
:The $35$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $625$, $676$, $729$, $784$, $841$, $900$, $961$, $1024$, $1089$, $1156$:
::$1225 = 35 \times 35$
:The $49$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$, $780$, $820$, $861$, $903$, $946$, $990$, $1035$, $1081$, $1128$, $1176$:
::$1225 = \ds \sum_{k \mathop = 1}^{49} k = \dfrac {49 \times \paren {49 + 1} } 2$
== Also see ==
* 
* 
* 
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== Sources ==
* : $1225$
* : $1225$
Category:Specific Numbers
Category:1225
1229

== Number ==
$1229$ (one thousand, two hundred and twenty-nine) is:
:The $201$st prime number
:The number of primes with no more than $4$ digits:
::$2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $\ldots$, $9907$, $9923$, $9929$, $9931$, $9941$, $9949$, $9967$, $9973$
:The smaller of the $42$nd pair of twin primes:
::$1231 - 1229 = 2$
:The $44$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $761$, $809$, $911$, $953$, $1013$, $1019$, $1031$, $1049$, $1103$, $1223$:
::$2 \times 1229 + 1 = 2459$, which is prime.
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1229
123

== Number ==
$123$ (one hundred and twenty-three) is:
:$3 \times 41$
:The index (after $2$, $3$, $6$, $30$, $75$, $81$, $115$) of the $8$th Woodall prime:
::$123 \times 2^{123} - 1$
:The $10$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$, $18$, $29$, $47$, $76$:
::$123 = 47 + 76$
:The $24$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:123
123,454,321

== Number ==
$123 \, 454 \, 321$ (one hundred and twenty-three million, four hundred and fifty-four thousand, three hundred and twenty-one) is:
:$41^2 \times 271^2$
:The $5$th Wonderful Demlo number after $1$, $121$, $12 \, 321$, $1 \, 234 \, 321$
:The $11 \, 111$th square number:
::$123 \, 454 \, 321 = 11 \, 111 \times 11 \, 111$
:A palindromic square of a palindromic number
== Also see ==
* Square of Small-Digit Palindromic Number is Palindromic/Examples/11111
* 
* 
Category:Specific Numbers
Category:123,454,321
123,455,554,321

== Number ==
$123 \, 455 \, 554 \, 321$ is:
:$11 \times 41 \times 73 \times 101 \times 137 \times 271$
:A palindromic Smith number:
::$1 + 2 + 3 + 4 + 5 + 5 + 5 + 5 + 4 + 3 + 2 + 1 = 40 = 1 + 1 + 4 + 1 + 7 + 3 + 1 + 0 + 1 + 1 + 3 + 7 + 2 + 7 + 1$
== Also see ==
* 
== Sources ==
* : $12,345,554,321$
Category:Specific Numbers
Category:123,455,554,321
123,456,789

== Number ==
$123 \, 456 \, 789$ is:
:$3^2 \times 3607 \times 3803$
:Remains penholodigital when multiplied by $2$, $4$, $5$, $7$ and $8$
== Also see ==
* Penholodigital Properties of 123,456,789
== Sources ==
* : $123,456,789$
* : $123,456,789$
Category:Specific Numbers
Category:123,456,789
1233

== Number ==
$1233$ (one thousand, two hundred and thirty-three) is:
:$3^2 \times 137$
:The sum of the squares of the parts when split into two:
::$1233 = 144 + 1089 = 12^2 + 33^2$
== Also see ==
* 
== Sources ==
* : $1233$
* : $1233$
Category:Specific Numbers
Category:1233
123456789 x 8 + 9 = 987654321

== Theorem ==
The above pattern is an instance of the identity:
:$\ds \paren {b - 2} \sum_{j \mathop = 1}^n j b^{n - j} + n = \sum_{j \mathop = 1}^n \paren {b - j} b^{n - j}$
where $b = 10$ and $n$ goes from $1$ to $9$.
== Proof ==
The proof proceeds by induction.
Let $n, b \in \Z_{>0}$, where $b \ge 3$.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:$\ds \paren {b - 2} \sum_{j \mathop = 1}^n j b^{n - j} + n = \sum_{j \mathop = 1}^n \paren {b - j} b^{n - j}$
=== Basis for the Induction ===
$\map P 1$ is the case:
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
:$\ds \paren {b - 2} \sum_{j \mathop = 1}^k j b^{k - j} + k = \sum_{j \mathop = 1}^k \paren {b - j} b^{k - j}$
from which it is to be shown that:
:$\ds \paren {b - 2} \sum_{j \mathop = 1}^{k + 1} j b^{k + 1 - j} + \paren {k + 1} = \sum_{j \mathop = 1}^{k + 1} \paren {b - j} b^{k + 1 - j}$
=== Induction Step ===
This is the induction step:
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
:$\ds \paren {b - 2} \sum_{j \mathop = 1}^n j b^{n - j} + n = \sum_{j \mathop = 1}^n \paren {b - j} b^{n - j}$
Category:Recreational Mathematics
123456789 x 9 + 10 = 1111111111

== Theorem ==
The above pattern is an instance of the identity:
:$\ds \paren {b - 1} \sum_{j \mathop = 1}^n j b^{n - j} + n + 1 = \sum_{j \mathop = 0}^n b^j$
where $b = 10$ and $n$ goes from $1$ to $9$.
== Proof ==
The proof proceeds by induction.
Let $n, b \in \Z_{>0}$, where $b \ge 3$.
For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:
:$\ds \paren {b - 1} \sum_{j \mathop = 1}^n j b^{n - j} + n + 1 = \sum_{j \mathop = 0}^n b^j$
=== Basis for the Induction ===
$\map P 1$ is the case:
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
:$\ds \paren {b - 1} \sum_{j \mathop = 1}^k j b^{k - j} + k + 1 = \sum_{j \mathop = 0}^k b^j$
from which it is to be shown that:
:$\ds \paren {b - 1} \sum_{j \mathop = 1}^{k + 1} j b^{k + 1 - j} + k + 2 = \sum_{j \mathop = 0}^{k + 1} b^j$
=== Induction Step ===
This is the induction step:
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
:$\ds \paren {b - 1} \sum_{j \mathop = 1}^n j b^{n - j} + n + 1 = \sum_{j \mathop = 0}^n b^j$
Category:Recreational Mathematics
124

== Number ==
$124$ (one hundred and twenty-four) is:
:$2^2 \times 31$
:The $2$nd Fermat pseudoprime to base $5$ after $4$:
::$5^{124} \equiv 5 \pmod {124}$
:The $3$rd even integer that cannot be expressed as the sum of $2$ prime numbers of which the smaller one is $3$, $5$ or $7$.
:The $7$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$.
:The $17$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$, $86$, $90$, $94$, $98$, $114$, $118$, $122$:
::$\nexists m \in \Z_{>0}: \map \phi m = 124$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $55$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $95$, $96$, $100$, $101$, $102$, $107$, $112$, $116$ which cannot be expressed as the sum of distinct pentagonal numbers.
== Also see ==
* 
* 
* 
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Category:Specific Numbers
Category:124
1240

== Number ==
$1240$ (one thousand, two hundred and forty) is:
:$2^3 \times 5 \times 31$
:The $15$th square pyramidal number after $1$, $5$, $14$, $30$, $55$, $91$, $140$, $204$, $285$, $385$, $506$, $650$, $819$, $1015$:
::$1240 = \ds \sum_{k \mathop = 1}^{15} k^2 = \dfrac {15 \paren {15 + 1} \paren {2 \times 15 + 1} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:1240
1247

== Number ==
$1247$ (one thousand, two hundred and forty-seven) is:
:$29 \times 43$
:The $11$th Fermat pseudoprime to base $4$ after $15$, $85$, $91$, $341$, $435$, $451$, $561$, $645$, $703$, $1105$:
::$4^{1247} \equiv 4 \pmod {1247}$
:The $29$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $477$, $532$, $590$, $651$, $715$, $782$, $852$, $925$, $1001$, $1080$, $1162$:
::$1247 = \ds \sum_{k \mathop = 1}^{29} \paren {3 k - 2} = \dfrac {29 \paren {3 \times 29 - 1} } 2$
:The $57$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $925$, $950$, $1001$, $1027$, $1080$, $1107$, $1162$, $1190$:
::$1247 = \ds \sum_{k \mathop = 1}^{29} \paren {3 k - 2} = \dfrac {29 \paren {3 \times 29 - 1} } 2$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1247
1249

== Number ==
$1249$ (one thousand, two hundred and forty-nine) is:
:The $204$th prime number
:The $6$th after $199$, $409$, $619$, $829$, $1039$ of the smallest $10$ primes in arithmetic sequence:
::$1249 = 5 \times 210 + 199$
:The smaller of the $23$rd pair of primes whose prime gap is $10$:
::$1259 - 1249 = 10$
:The $26$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $\ldots$, $501$, $624$, $625$, $749$, $751$, $875$, $999$:
::$1249^3 = 1 \, 948 \, 44 \mathbf {1 \, 249}$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1249
125

== Number ==
$125$ (one hundred and twenty-five) is:
:$5^3$
:The length of the shortest face diagonal of the smallest cuboid whose edges and the diagonals of whose faces are all integers:
::The lengths of the edges are $44$, $117$, $240$
::The lengths of the diagonals of the faces are $125$, $244$, $267$.
:The $3$rd power of $5$ after $(1)$, $5$, $25$:
::$125 = 5^3$
:The $3$rd Friedman number base $10$ after $25$, $121$:
::$125 = 5^{1 + 2}$
:The $4$th positive integer after $50$, $65$, $85$, and first cube, which can be expressed as the sum of two square numbers in two or more different ways:
::$125 = 11^2 + 2^2 = 10^2 + 5^2 = 5^3$
:The $5$th cube number after $1$, $8$, $27$, $64$:
::$125 = 5 \times 5 \times 5$
:The $10$th integer after $7$, $13$, $19$, $35$, $38$, $41$, $57$, $65$, $70$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$125^2 = 15 \, 625$; $1 = 1^2$, $5625 = 75^2$
:The $13$th trimorphic number after $1$, $4$, $5$, $6$, $9$, $24$, $25$, $49$, $51$, $75$, $76$, $99$:
::$125^3 = 1 \, 953 \, \mathbf {125}$
:The $13$th positive integer $n$ after $0$, $1$, $5$, $25$, $29$, $41$, $49$, $61$, $65$, $85$, $89$, $101$ such that the Fibonacci number $F_n$ ends in $n$
:The $17$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$, $100$, $108$, $121$
:The $57$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $95$, $96$, $102$, $108$, $114$, $119$, $120$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* Cuboid with Integer Edges and Face Diagonals
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== Sources ==
* : $125$
* : $125$
Category:Specific Numbers
Category:125
125,000

== Number ==
$125 \, 000$ (one hundred and twenty-five thousand) is:
:$2^3 \times 5^6$
:The $50$th cube number
::$125 \, 000 = 50 \times 50 \times 50$
== Also see ==
* 
== Historical Note ==
== Sources ==
* : $125,000$
Category:Specific Numbers
Category:125,000
125,000/Historical Note

== Historical Note on $125\,000$ (One Hundred and Twenty-Five Thousand) ==
The second case of Fermat's Last Theorem, where the value $p$ in the equation $x^p + y^p = z^p$ divides one of $x$, $y$ and $z$, had been proved impossible for all values of $p$ up to $125\,000$, at the point at which finally proved it in $1994$.
== Sources ==
* : $125,000$
Category:Historical Notes
125,411,328,000

== Number ==
$125 \, 411 \, 328 \, 000$ is:
:$2^{16} \times 3^7 \times 5^3 \times 7$
:The $7$th superfactorial after $1$, $2$, $12$, $288$, $34 \, 560$, $24 \, 883 \, 200$:
::$125 \, 411 \, 328 \, 000 = 7\$ = 7! \times 6! \times 5! \times 4! \times 3! \times 2! \times 1!$
== Also see ==
* 
Category:Specific Numbers
Category:125,411,328,000
1255

== Number ==
$1255$ (one thousand, two hundred and fifty-five) is:
:$5 \times 251$
:The $14$th non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $58$, $85$, $91$, $130$, $214$, $226$, $370$, $526$, $706$, $730$, $771$, $1255$, $\ldots$
:The number of integer partitions for $23$:
::$\map p {23} = 1255$
:The $31$st positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $706$, $730$, $771$, $784$, $841$, $1024$, $1089$, $1225$, $1255$, $\ldots$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1255
126

== Number ==
$126$ (one hundred and twenty-six) is:
:$2 \times 3^2 \times 7$
:The $4$th even integer after $98$, $122$, $124$ that cannot be expressed as the sum of $2$ prime numbers of which the smaller one is $3$, $5$ or $7$.
:The $4$th Friedman number base $10$ after $25$, $121$, $125$:
::$126 = 21 \times 6$
:The $6$th pentatope number after $1$, $5$, $15$, $35$, $70$:
::$126 = 1 + 4 + 10 + 20 + 35 + 56 = \dfrac {6 \paren {6 + 1} \paren {6 + 2} \paren {6 + 3} } {24}$
:The $6$th pentagonal pyramidal number after $1$, $6$, $12$, $40$, $75$:
::$126 = 1 + 5 + 12 + 22 + 35 + 51 = \dfrac {6^2 \paren {6 + 1} } 2$
:The $9$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$, $100$:
::$126 = \dfrac {9 \paren {3 \times 9 + 1} } 2$
:The $18$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$, $51$, $57$, $70$, $77$, $92$, $100$, $117$:
::$126 = \dfrac {9 \paren {3 \times 9 + 1} } 2$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:126
1260

== Number ==
$1260$ (one thousand, two hundred and sixty) is:
:$2^2 \times 3^2 \times 5 \times 7$
:The $16$th highly composite number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$:
::$\map {\sigma_0} {1260} = 36$
:The $16$th superabundant number after $1$, $2$, $4$, $6$, $12$, $24$, $36$, $48$, $60$, $120$, $180$, $240$, $360$, $720$, $840$:
::$\dfrac {\map {\sigma_1} {1260} } {1260} = \dfrac {4368} {1260} = 3 \cdotp 4 \dot 6$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1260
1261

== Number ==
$1261$ (one thousand, two hundred and sixty-one) is:
:$13 \times 97$
:The $21$st centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$, $397$, $469$, $547$, $631$, $721$, $817$, $919$, $1027$, $1141$:
::$1261 = \ds 1 + \sum_{k \mathop = 1}^{21 - 1} 6 k = 21^3 - 20^3$
:The $4$th integer after $3$, $228$, $671$ whose square is the sum of $2$ coprime cubes:
::$1261^2 = ??^3 + ??^3$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1261
1264

== Number ==
$1264$ (one thousand, two hundred and sixty-four) is:
:$2^4 \times 79$
:The $24$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $904$, $906$, $908$, $1114$, $1116$, $1118$, $1144$, $1146$, $1148$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes.
== Also see ==
* 
Category:Specific Numbers
Category:1264
1266

== Number ==
$1266$ (one thousand, two hundred and sixty-six) is:
:$2 \times 3 \times 211$
:The $25$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $904$, $906$, $908$, $1114$, $1116$, $1118$, $1144$, $1146$, $1148$, $1264$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes.
== Also see ==
* 
Category:Specific Numbers
Category:1266
1268

== Number ==
$1268$ (one thousand, two hundred and sixty-eight) is:
:$2^2 \times 317$
:The $26$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $1148$, $1264$, $1266$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
== Also see ==
* 
Category:Specific Numbers
Category:1268
127

== Number ==
$127$ (one hundred and twenty-seven) is:
:The $31$st prime number
:The larger of the $1$st pair of primes whose prime gap is $14$:
::$127 - 113 = 14$
:The $2$nd obstinate number after $1$.
:The $4$th Mersenne number and $4$th Mersenne prime after $3$, $7$, $31$:
::$127 = 2^7 - 1$
:The $5$th Friedman number base $10$ after $25$, $121$, $125$, $126$:
::$127 = 2^7 − 1$
:The $7$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$:
::$127 = 1 + 6 + 12 + 18 + 24 + 30 + 36 = 7^3 - 6^3$
:The $8$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 7^2 + 29 = 127$ ()
:The index of the $11$th Mersenne number after $1$, $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $67$ which asserted to be prime
::(note that he missed $89$ and $107$)
:The index of the $12$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$, $107$:
::$M_{127} = 2^{127} - 1 = 170 \, 141 \, 183 \, 460 \, 469 \, 231 \, 731 \, 687 \, 303 \, 715 \, 884 \, 105 \, 727$
:The $26$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $75$, $79$, $87$, $93$, $99$, $105$, $111$, $115$, $127$, $\ldots$
== Also see ==
* 
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== Sources ==
* : $127$
* : $127$
Category:Specific Numbers
Category:127
1275

== Number ==
$1275$ (one thousand, two hundred and seventy-five) is:
:$3 \times 5^2 \times 17$
:The $50$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $703$, $741$, $780$, $820$, $861$, $903$, $946$, $990$, $1035$, $1081$, $1128$, $1176$, $1225$:
::$1275 = \ds \sum_{k \mathop = 1}^{50} k = \dfrac {50 \times \paren {50 + 1} } 2$
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Category:Specific Numbers
Category:1275
1276

== Number ==
$1276$ (one thousand, two hundred and seventy-six) is:
:$2^2 \times 11 \times 29$
:The $29$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $610$, $672$, $737$, $805$, $876$, $950$, $1027$, $1107$, $1190$:
::$1276 = \dfrac {28 \paren {3 \times 29 + 1} } 2$
:The $58$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $925$, $950$, $1001$, $1027$, $1080$, $1107$, $1162$, $1190$, $1247$:
::$1276 = \dfrac {28 \paren {3 \times 29 + 1} } 2$
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Category:Specific Numbers
Category:1276
1279

== Number ==
$1279$ (one thousand, two hundred and seventy-nine) is:
:The $207$th prime number
:The index of the $15$th Mersenne prime after $2$, $3$, $5$, $7$, $13$, $17$, $19$, $31$, $61$, $89$, $107$, $127$, $521$, $607$:
::$M_{1279} = 2^{1279} - 1 \approx 1 \cdotp 041 \times 10^{385}$
:The $26$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 25^2 + 29 = 1279$ ()
:The larger of the $43$rd pair of twin primes:
::$1279 - 1277 = 2$
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Category:Specific Numbers
Category:1279
128

== Number ==
$128$ (one hundred and twenty-eight) is:
:$2^7$
:In binary:
::$128_{10} = 10 \, 000 \, 000_2$
:The $1$st positive integer with $7$ or more prime factors:
::$128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
:The $2$nd seventh power after $1$:
::$128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
:The $3$rd positive integer after $64$, $96$ with $6$ or more prime factors:
::$128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \paren {\mathop \times 2}$
:The $5$th and last of the known powers of $2$ whose digits are also all powers of $2$:
::$1$, $2$, $4$, $8$, $128$
:The $6$th Friedman number base $10$ after $25$, $121$, $125$, $126$, $127$:
::$128 = 2^{8 − 1}$
:The $7$th power of $2$ after $(1)$, $2$, $4$, $8$, $16$, $32$, $64$:
::$128 = 2^7$
:The $8$th almost perfect number after $1$, $2$, $4$, $8$, $16$, $32$, $64$:
::$\map {\sigma_1} {128} = 255 = 2 \times 128 - 1$
:The $18$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$, $115$:
::$128 = 8 \times 16 = 8 \times \paren {1 \times 2 \times 8}$
:The $19$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$, $100$, $108$, $121$, $125$
:The $31$st and largest positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $67$, $72$, $76$, $92$, $96$, $108$, $112$, $128$
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== Sources ==
* : $128$
* : $128$
Category:Specific Numbers
Category:128
1281

== Number ==
$1281$ (one thousand, two hundred and eighty-one) is:
:$3 \times 7 \times 61$
:The $21$st octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $645$, $736$, $833$, $936$, $1045$, $1160$:
::$1281 = \ds \sum_{k \mathop = 1}^{21} \paren {6 k - 5} = 21 \paren {3 \times 21 - 2}$
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Category:Specific Numbers
Category:1281
1283

== Number ==
$1283$ (one thousand, two hundred and eighty-three) is:
:The $208$th prime number
:The $28$th safe prime after $5$, $7$, $11$, $23$, $47$, $\ldots$, $479$, $503$, $563$, $587$, $719$, $839$, $863$, $887$, $983$, $1019$, $1187$:
::$1283 = 2 \times 553 + 1$
:The $56$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $743$, $773$, $797$, $823$, $853$, $883$, $937$, $947$, $953$, $967$, $983$, $997$, $1223$
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Category:Specific Numbers
Category:1283
1288

== Number ==
$1288$ (one thousand, two hundred and eighty-eight) is:
:$2^3 \times 7 \times 23$
:The $23$rd heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$, $342$, $403$, $469$, $540$, $616$, $697$, $783$, $874$, $970$, $1071$, $1177$:
::$1288 = \ds \sum_{k \mathop = 1}^{23} \paren {5 k - 4} = \dfrac {23 \paren {5 \times 23 - 3} } 2$
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Category:Specific Numbers
Category:1288