nyuuzyou/wikis huggingface dataset:
111,111,111,111

== Number ==
$111 \, 111 \, 111 \, 111$ is:
:$3 \times 7 \times 11 \times 13 \times 37 \times 101 \times 9901$
:The $12$th repunit
 
== Also see ==
* 
Category:Specific Numbers
Category:111,111,111,111
111,111,111,111,111

== Number ==
$111 \, 111 \, 111 \, 111 \, 111$ is:
:$3 \times 31 \times 37 \times 41 \times 271 \times 2 \, 906 \, 161$
:The $15$th repunit
 
== Also see ==
* 
Category:Specific Numbers
Category:111,111,111,111,111
111,111,111,111,111,111

== Number ==
$111 \, 111 \, 111 \, 111 \, 111 \, 111$ is:
:$3^2 \times 7 \times 11 \times 13 \times 19 \times 37 \times 52 \, 579 \times 333 \, 667$
:The $18$th repunit
 
== Also see ==
* 
Category:Specific Numbers
Category:111,111,111,111,111,111
111,111,111,111,111,111,111

== Number ==
$111 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$ is:
:$3 \times 37 \times 43 \times 239 \times 1933 \times 4649 \times 10 \, 838 \, 689$
:The $21$st repunit
== Also see ==
* 
Category:Specific Numbers
Category:111,111,111,111,111,111,111
111,777

== Number ==
$111 \, 777$ (one hundred and eleven thousand, seven hundred and seventy-seven) is:
:$3 \times 19 \times 37 \times 53$
:The least integer not nameable in fewer than nineteen syllables (which, paradoxically, is itself an expression of eighteen syllables)
== Also see ==
* Berry's Paradox
== Sources ==
* : $111,777$
* 
* 
Category:Specific Numbers
Category:111,777
1111

== Number ==
$1111$ (one thousand, one hundred and eleven) is:
:$11 \times 101$
:The $4$th repuint after $1$, $11$, $111$
:The $8$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $77$, $363$ which is the index of a palindromic triangular number
::$T_{1111} = 617 \, 716$
:The $13$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $111$, $121$, $202$, $212$, $1001$ whose square is also palindromic integer
::$1111^2 = 1 \, 234 \, 321$
:The $35$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $216$, $224$, $312$, $315$, $384$, $432$, $612$, $624$, $672$, $735$, $816$:
::$1111 = 1111 \times 1 = 1111 \times \paren {1 \times 1 \times 1 \times 1}$
:The $51$st Smith number after $4$, $22$, $27$, $58$, $\ldots$, $778$, $825$, $852$, $895$, $913$, $915$, $922$, $958$, $985$, $1086$:
::$1 + 1 + 1 + 1 = 1 + 1 + 1 + 0 + 1 = 4$
:$1111 = 56^2 - 45^2$
== Also see ==
* Difference between Two Squares equal to Repunit
* 
* 
* 
* 
* 
== Sources ==
* : $1111$
* : $1111$
Category:Specific Numbers
Category:1111
1112

== Number ==
$1112$ (one thousand one hundred and twelve) is:
:$2^3 \times 139$
:The $36$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $432$, $612$, $624$, $672$, $735$, $816$, $1111$:
::$1112 = 556 \times 2 = 556 \times \paren {1 \times 1 \times 1 \times 2}$
== Also see ==
* 
Category:Specific Numbers
Category:1112
1113

== Number ==
$1113$ (one thousand one hundred and thirteen) is:
:$3 \times 7 \times 53$
:The $37$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $432$, $612$, $624$, $672$, $735$, $816$, $1111$, $1112$:
::$1113 = 371 \times 3 = 371 \times \left({1 \times 1 \times 1 \times 3}\right)$
== Also see ==
* 
Category:Specific Numbers
Category:1113
1114

== Number ==
$1114$ (one thousand, one hundred and fourteen) is:
:$2 \times 557$
:The $18$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $904$, $906$, $908$ 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:1114
1115

== Number ==
$1115$ (one thousand one hundred and fifteen) is:
:$5 \times 223$
:The $38$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $432$, $612$, $624$, $672$, $735$, $816$, $1111$, $1112$, $1113$:
::$1115 = 223 \times 5 = 223 \times \left({1 \times 1 \times 1 \times 5}\right)$
== Also see ==
* 
Category:Specific Numbers
Category:1115
1116

== Number ==
$1116$ (one thousand, one hundred and sixteen) is:
:$2^2 \times 3^2 \times 31$
:The $19$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $904$, $906$, $908$, $1114$ that cannot be expressed as the sum of $2$ prime numbers which are each one of a pair of twin primes
:The $39$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $432$, $612$, $624$, $672$, $735$, $816$, $1111$, $1112$, $1113$, $1115$:
::$1116 = 186 \times 6 = 186 \times \paren {1 \times 1 \times 1 \times 6}$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1116
1117

== Number ==
$1117$ (one thousand, one hundred and seventeen) is:
:The $187$th prime number
:The larger of the $16$th pair of primes whose prime gap is $8$:
::$1117 - 1109 = 8$
:The $47$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $877$, $881$, $883$, $887$, $911$, $919$, $929$, $977$, $991$, $997$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1117
1118

== Number ==
$1118$ (one thousand, one hundred and eighteen) is:
:$2 \times 13 \times 43$
:The $20$th even integer after $2$, $4$, $94$, $96$, $\ldots$, $904$, $906$, $908$, $1114$, $1116$ 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:1118
112

== Number ==
$112$ (one hundred and twelve) is:
:$2^4 \times 7$
:The $7$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$:
::$112 = 1 + 7 + 11 + 16 + 21 + 26 + 31 = \dfrac {7 \left({5 \times 7 - 3}\right)} 2$
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $8$ different ways
:The $16$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$:
::$112 = 56 \times 2 = 56 \times \left({1 \times 1 \times 2}\right)$
:The $30$th positive integer which is not the sum of $1$ or more distinct squares:
::$2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $19$, $22$, $23$, $24$, $27$, $28$, $31$, $32$, $33$, $43$, $44$, $47$, $48$, $60$, $67$, $72$, $76$, $92$, $96$, $108$, $112$, $\ldots$
:The $53$rd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $95$, $96$, $100$, $101$, $102$, $107$ which cannot be expressed as the sum of distinct pentagonal numbers
:The length of the side of the smallest perfect square dissection of an integer square
:The side length of the smallest equilateral triangle with sides of integer length which contains a point which is an integer distance from each vertex
== Historical Note ==
== Also see ==
* Smallest Perfect Square Dissection
* Smallest Equilateral Triangle with Internal Point at Integer Distances from Vertices
* 
* 
* 
* 
* 
== Sources ==
* : $112$
* : $112$
Category:Specific Numbers
Category:112
112,359,550,561,797,752,809

== Number ==
$112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 809$ is:
:$101 \times 1 \, 052 \, 788 \, 969 \times 1 \, 056 \, 689 \, 261$
:The smallest positive integer which is multiplied by $99$ when $1$ is appended to each end.
== Also see ==
* Smallest Number which is Multiplied by 99 by Appending 1 to Each End
== Sources ==
* : $112,359,550,561,797,732,809$
:::but note the transcription error in the $5$th digit from the end: it should be $5$ not $3$.
Category:Specific Numbers
Category:112,359,550,561,797,752,809
1123

== Number ==
$1123$ (one thousand, one hundred and twenty-three) is:
:The $186$th prime number
:The $17$th balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$, $563$, $593$, $607$, $653$, $733$, $947$, $977$, $1103$:
::$1123 = \dfrac {1117 + 1129} 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1123
1127

== Number ==
$1127$ (one thousand, one hundred and twenty-seven) is:
:$7^2 \times 23$
:The $1$st number whose square consists of $2$ consecutive odd or even integers juxtaposed:
::$1127^2 = 01 \, 270 \, 129$
== Also see ==
* 
== Sources ==
* : $1127$
* : $1127$
Category:Specific Numbers
Category:1127
1128

== Number ==
$1128$ (one thousand, one hundred and twenty-eight) is:
:$2^3 \times 3 \times 47$
:The $24$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $703$, $780$, $861$, $946$, $1035$:
::$1128 = \ds \sum_{k \mathop = 1}^{24} \paren {4 k - 3} = 24 \paren {2 \times 24 - 1}$
:The $47$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $496$, $528$, $561$, $595$, $630$, $666$, $703$, $741$, $780$, $820$, $861$, $903$, $946$, $990$, $1035$, $1081$:
::$1128 = \ds \sum_{k \mathop = 1}^{47} k = \dfrac {47 \times \paren {47 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1128
1129

== Number ==
$1129$ (one thousand, one hundred and twenty-nine) is:
:The $189$th prime number
:The $2$nd prime number after $1009$ which can be expressed in the form $x^2 + n y^2$ for all values of $n$ from $1$ to $10$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1129
113

== Number ==
$113$ (one hundred and thirteen) is:
:The $30$th prime number
:The $1$st $3$-digit permutable prime:
::$113$, $131$, $311$
:The smaller of the $1$st pair of primes whose prime gap is $14$:
::$127 - 113 = 14$
:The $2$nd near-repdigit prime after $101$
:The $7$th Proth prime after $3$, $5$, $13$, $17$, $41$, $97$:
::$113 = 7 \times 2^4 + 1$
:The $10$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$
:The $11$th Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $53$, $83$, $89$:
::$2 \times 113 + 1 = 227$, which is prime.
:The $11$th long period prime after $7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$
:The $14$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$
:The $16$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$
== Also see ==
* Smallest 3-Digit Permutable Prime
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $113$
* : $113$
Category:Specific Numbers
Category:113
1131

== Number ==
$1131$ (one thousand, one hundred and thirty-one) is:
:$3 \times 13 \times 29$
:The $40$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $432$, $612$, $624$, $672$, $735$, $816$, $1111$, $1112$, $1113$, $1115$, $1116$:
::$1131 = 377 \times 3 = 377 \times \left({1 \times 1 \times 3 \times 1}\right)$
== Also see ==
* 
Category:Specific Numbers
Category:1131
114

== Number ==
$114$ (one hundred and fourteen) is:
:$2 \times 3 \times 19$
:The number of different ways to colour the faces of a cube with $3$ given colours, one colour per face.
:The $9$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$:
::$114 = 2 \times 3 \times 19$
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $10$ ways.
:The $14$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$, $86$, $90$, $94$, $98$:
::$\nexists m \in \Z_{>0}: \map \phi m = 114$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $54$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $77$, $78$, $79$, $84$, $90$, $91$, $95$, $96$, $102$, $108$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* Number of Different Ways to Colour the Faces of Cube with 3 Colours‎
* 
* 
* 
* 
== Sources ==
* : $114$
Category:Specific Numbers
Category:114
114,689

== Number ==
$114 \, 689$ is:
:The $10 \, 847$th prime.
:One of the prime factors of the $13$th Fermat number $2^{\paren {2^{12} } } + 1$:
::$114 \, 689 = 7 \times 2^{14} + 1$
== Also see ==
* 
== Sources ==
* : $257$
* : $257$
Category:Specific Numbers
Category:114,689
114, 381,625, 757,888, 867,669, 235,779, 976,146, 612,010, 218,296, 721,242, 362,562, 561,842, 935,706, 935,245, 733,897, 830,597, 123,563, 958,705, 058,989, 075,147, 599,290, 026,879, 543,541

== Number ==
$114 \, 381 \, 625 \, 757 \, 888 \, 867 \, 669 \, 235 \, 779 \, 976 \, 146 \, 612 \, 010 \, 218 \, 296 \, 721 \, 242 \, 362 \, 562 \, 561 \, 842 \, 935 \, 706 \, 935 \, 245 \, 733 \, 897 \, 830 \, 597 \, 123 \, 563 \, 958 \, 705 \, 058 \, 989 \, 075 \, 147 \, 599 \, 290 \, 026 \, 879 \, 543 \, 541$ is:
:Known as RSA $129$, constructed as a cryptographic challenge by , and .
== Sources ==
* : $114,381,625,757,888,867,669,235,779,976,146,612,010,218,296,721,242,362,562,561,842,935,706,935,245,733,897,830,597,123,563,958,705,058,989,075,147,599,290,026,879,543,541$
Category:Specific Numbers
Category:RSA 129
1140

== Number ==
$1140$ (one thousand, one hundred and forty) is:
:$2^2 \times 3 \times 5 \times 19$
:The $18$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$, $220$, $286$, $364$, $455$, $560$, $680$, $816$, $969$:
::$1140 = \ds \sum_{k \mathop = 1}^{18} \frac {k \paren {k + 1} } 2 = \dfrac {18 \paren {18 + 1} \paren {18 + 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:1140
1141

== Number ==
$1141$ (one thousand, one hundred and forty-one) is:
:$7 \times 163$
:The $20$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $169$, $217$, $271$, $331$, $397$, $469$, $547$, $631$, $721$, $817$, $919$, $1027$:
::$1141 = \ds 1 + \sum_{k \mathop = 1}^{20 - 1} 6 k = 20^3 - 19^3$
:The smallest known integer whose $6$th power can be expressed as the sum of $7$ smaller $6$th powers:
::$1141^6 = 74^6 + 234^6 + 402^6 + 474^6 + 702^6 + 894^6 + 1077^6$
== Also see ==
* Sixth Power as Sum of 7 Sixth Powers
* 
== Sources ==
* : $1141$
* : $1141$
Category:Specific Numbers
Category:1141
1144

== Number ==
$1144$ (one thousand, one hundred and forty-four) is:
:$2^3 \times 11 \times 13$
:The $21$st even integer after $2$, $4$, $94$, $96$, $\ldots$, $904$, $906$, $908$, $1114$, $1116$, $1118$ 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:1144
1146

== Number ==
$1146$ (one thousand, one hundred and forty-six) is:
:$2 \times 3 \times 191$
:The $22$nd even integer after $2$, $4$, $94$, $96$, $\ldots$, $904$, $906$, $908$, $1114$, $1116$, $1118$, $1144$ 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:1146
1148

== Number ==
$1148$ (one thousand, one hundred and forty-eight) is:
:$2^2 \times 7 \times 41$
:The $5$th positive integer after $504$, $756$, $806$, $1008$ which can be expressed as the product of $2$ two-digit numbers in $2$ ways such that the factors in one of those pairs is the reversal of each of the factors in the other:
::$1148 = 14 \times 82 = 28 \times 41$
:The $23$rd even integer after $2$, $4$, $94$, $96$, $\ldots$, $904$, $906$, $908$, $1114$, $1116$, $1118$, $1144$, $1146$ 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:1148
115

== Number ==
$115$ (one hundred and fifteen) is:
:$5 \times 23$
:The $1$st term of the $2$nd $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
::$\tuple {115, 116, 117, 118, 119}$
:The $5$th heptagonal pyramidal number after $1$, $8$, $26$, $60$:
::$115 = 1 + 7 + 18 + 34 + 55 = \dfrac {5 \paren {5 + 1} \paren {5 \times 5 - 2} } 6$
:The index (after $2$, $3$, $6$, $30$, $75$, $81$) of the $7$th Woodall prime:
::$115 \times 2^{115} - 1$
:The $8$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$ whose divisor sum is square:
:
:The $17$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $15$, $24$, $36$, $111$, $112$:
::$115 = 23 \times 5 = 23 \times \paren {1 \times 1 \times 5}$
:The $25$th lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $43$, $49$, $51$, $63$, $67$, $73$, $75$, $79$, $87$, $93$, $99$, $105$, $111$, $115$, $\ldots$
:The $36$th semiprime:
::$115 = 5 \times 23$
:The $38$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $89$, $97$, $101$, $103$, $115$, $\ldots$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:115
115,132,219,018,763,992,565,095,597,973,971,522,400

== Number ==
$115 \, 132 \, 219 \, 018 \, 763 \, 992 \, 565 \, 095 \, 597 \, 973 \, 971 \, 522 \, 400$ is:
:$2^5 \times 5^2 \times 19 \times 2237 \times 102 \, 351 \, 435 \, 827 \times 33 \, 082 \, 122 \, 361 \, 353 \, 352 \, 663$
:The $87$th pluperfect digital invariant :
== Also see ==
* 
Category:Specific Numbers
Category:115,132,219,018,763,992,565,095,597,973,971,522,400
115,132,219,018,763,992,565,095,597,973,971,522,401

== Number ==
$115 \, 132 \, 219 \, 018 \, 763 \, 992 \, 565 \, 095 \, 597 \, 973 \, 971 \, 522 \, 401$ is:
:$32 \times 17 \, 669 \times 84 \, 522 \, 233 \times 8 \, 565 \, 869 \, 088 \, 228 \, 936 \, 598 \, 488 \, 557$
:The $88$th and largest pluperfect digital invariant:
== Also see ==
* 
== Sources ==
* : $115,132,219,018,763,992,565,095,597,973,971,522,401$
* : $115,132,219,018,763,992,565,095,597,973,971,522,401$
Category:Specific Numbers
Category:115,132,219,018,763,992,565,095,597,973,971,522,401
1151

== Number ==
$1151$ (one thousand one hundred and fifty-one) is:
:The $190$th prime number
:The smaller of the $41$st pair of twin primes:
::$1153 - 1151 = 2$
:The $44$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$, $1033$, $1061$, $1069$, $1091$, $1097$, $1103$, $1109$
:The $48$th near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $877$, $881$, $883$, $887$, $911$, $919$, $929$, $977$, $991$, $997$, $1117$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1151
1153

== Number ==
$1153$ (one thousand one hundred and fifty-three) is:
:The $191$st prime number
:The $18$th Proth prime after $3$, $5$, $13$, $17$, $41$, $\ldots$, $449$, $577$, $641$, $673$, $769$, $929$:
::$1153 = 9 \times 2^7 + 1$
:The smaller of the $20$th pair of primes whose prime gap is $10$:
::$1163 - 1153 = 10$
:The larger of the $41$st pair of twin primes:
::$1153 - 1151 = 2$
:The $45$th emirp after $13$, $17$, $31$, $37$, $\ldots$, $941$, $953$, $967$, $971$, $983$, $991$, $1009$, $1021$, $1031$, $1033$, $1061$, $1069$, $1091$, $1097$, $1103$, $1109$, $1151$
== Also see ==
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:1153
1156

== Number ==
$1156$ (one thousand, one hundred and fifty-six) is:
:$2^2 \times 17^2$
:The $34$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $625$, $676$, $729$, $784$, $841$, $900$, $961$, $1024$, $1089$:
::$1156 = 34 \times 34$
== Also see ==
* 
Category:Specific Numbers
Category:1156
116

== Number ==
$116$ (one hundred and sixteen) is:
:$2^2 \times 29$
:The $2$nd term of the $2$nd $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
::$\tuple {115, 116, 117, 118, 119}$
:The $1$st of the $4$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {116} = \map {\sigma_0} {117} = 6$
:The $9$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 116$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $11$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$
:The $22$nd positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
:The $54$th positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $95$, $96$, $100$, $101$, $102$, $107$, $112$ which cannot be expressed as the sum of distinct pentagonal numbers.
== Also see ==
* 
* 
* 
* 
* 
* 
== Sources ==
* : $116$
Category:Specific Numbers
Category:116
116,415,321,826,934,814,453,125

== Number ==
$116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$ is:
:$5^{33}$
:The $48 \, 828 \, 125$th cube number:
::$116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125 = 48 \, 828 \, 125 \times 48 \, 828 \, 125 \times 48 \, 828 \, 125$
:The $33$rd power of $5$:
::$116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125 = 5^{33}$
:The larger divisor of the $10$th and largest power of $10$ which can be expressed as the product of $2$ factors neither of which has a zero in its decimal representation:
::$10^{33} = 8 \, 589 \, 934 \, 592 \times 116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$
== Also see ==
* Powers of 10 Expressible as Product of 2 Zero-Free Factors
* 
* 
== Sources ==
* : $10^{33}$
Category:Specific Numbers
Category:116,415,321,826,934,814,453,125
1160

== Number ==
$1160$ (one thousand, one hundred and sixty) is:
:$2^3 \times 5 \times 29$
:The $20$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$, $225$, $280$, $341$, $408$, $481$, $560$, $645$, $736$, $833$, $936$, $1045$:
::$1160 = \ds \sum_{k \mathop = 1}^{20} \paren {6 k - 5} = 20 \paren {3 \times 20 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:1160
1162

== Number ==
$1162$ (one thousand, one hundred and sixty-two) is:
:$2 \times 7 \times 83$
:The $28$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $477$, $532$, $590$, $651$, $715$, $782$, $852$, $925$, $1001$, $1080$:
::$1162 = \ds \sum_{k \mathop = 1}^{28} \paren {3 k - 2} = \dfrac {28 \paren {3 \times 28 - 1} } 2$
:The $55$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $610$, $651$, $672$, $715$, $737$, $782$, $805$, $852$, $876$, $925$, $950$, $1001$, $1027$, $1080$, $1107$:
::$1162 = \ds \sum_{k \mathop = 1}^{28} \paren {3 k - 2} = \dfrac {28 \paren {3 \times 28 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1162
1163

== Number ==
$1163$ (one thousand one hundred and sixty-three) is:
:The $192$nd prime number
:The smaller of the $17$th pair of primes whose prime gap is $8$:
::$1171 - 1163 = 8$
:The larger of the $20$th pair of primes whose prime gap is $10$:
::$1163 - 1153 = 10$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1163
1165

== Number ==
$1165$ (one thousand, one hundred and sixty-five) is:
:$5 \times 233$
:The $52$nd Smith number after $4$, $22$, $27$, $58$, $\ldots$, $778$, $825$, $852$, $895$, $913$, $915$, $922$, $958$, $985$, $1086$, $1111$:
::$1 + 1 + 6 + 5 = 5 + 2 + 3 + 3 = 13$
== Also see ==
* 
::No further elements of this sequence will be tabulated on .
Category:Specific Numbers
Category:1165
1166

== Number ==
$1166$ (one thousand, one hundred and sixty-six) is:
:$2 \times 11 \times 53$
:The $11$th heptagonal pyramidal number after $1$, $8$, $26$, $60$, $115$, $196$, $308$, $456$, $645$, $880$:
::$1166 = 1 + 7 + 18 + 34 + 55 + 81 + 112 + 148 + 189 + 235 + 286 = \dfrac {11 \paren {11 + 1} \paren {5 \times 11 - 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:1166
117

== Number ==
$117$ (one hundred and seventeen) is:
:$3^2 \times 13$
:The length of the $2$nd longest edge 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 term of the $2$nd $5$-tuple of consecutive integers have the property that they are not values of the divisor sum function $\map {\sigma_1} n$ for any $n$:
::$\tuple {115, 116, 117, 118, 119}$
:The $2$nd of the $4$th pair of consecutive integers which both have $6$ divisors:
::$\map {\sigma_0} {116} = \map {\sigma_0} {117} = 6$
:The $5$th of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
::$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$
:The $9$th pentagonal number after $1$, $5$, $12$, $22$, $35$, $51$, $70$, $92$:
::$117 = 1 + 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 = \dfrac {9 \paren {3 \times 9 - 1} } 2$
:The $17$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$, $51$, $57$, $70$, $77$, $92$, $100$:
::$117 = \dfrac {9 \paren {3 \times 9 - 1} } 2$
== Also see ==
* Cuboid with Integer Edges and Face Diagonals
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:117