nyuuzyou/wikis huggingface dataset:
333,667 is Unique Period Prime with Period 9

== Theorem ==
$333 \, 667$ is a unique period prime whose reciprocal has a period of $9$:
:$\dfrac 1 {333 \, 667} = 0 \cdotp \dot 00000 \, 299 \dot 7$
== Proof ==
By long division:
 0.000002997000002...
 ---------------------
333667)1.000000000000000000
 667334
 --------
 3326660
 3003003
 -------
 3236570
 3003003
 -------
 2335670
 2335669
 -------
 1000000
 667334
 -------
 ......
It remains to be shown that $333 \, 667$ is the only prime number whose reciprocal has a period of $9$.
We have that:
:$333 \, 667 \nmid 10$
From Period of Reciprocal of Prime, the period of such a prime is the order of $10$ modulo $p$.
That is, the smallest integer $d$ such that:
:$10^d \equiv 1 \pmod p$
From the above long division we know that the period of $\dfrac 1 {333 \, 667}$ is $9$, so $10^9 \equiv 1 \pmod {333 \, 667}$.
The only other possible primes $p$ whose reciprocals might have a period of $9$ must also satisfy:
:$10^9 \equiv 1 \pmod p$
that is:
:$p \divides 10^9 - 1$
Consider:
Therefore the only other possible primes whose reciprocals might have a period of $9$ are $3$ and $37$.
From Reciprocal of $37$:
:$\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$
and trivially:
:$\dfrac 1 3 = 0 \cdotp \dot 3$
which has a period of $1$.
As required, the only prime number whose reciprocal has a period of $9$ is $333 \, 667$.
== Sources ==
* : $333,667$
Category:333,667
Category:Examples of Unique Period Primes
3333

== Number ==
$3003$ (three thousand, three hundred and thirty-three) is:
:$3 \times 11 \times 101$
:$3333 = 67^2 - 34^2$
== Also see ==
* Difference between Two Squares equal to Repdigit
== Historical Note ==
== Sources ==
* : $3333$
Category:Specific Numbers
Category:3333
3333/Historical Note

== Historical Note on $3333$: (Three Thousand, Three Hundred and Thirty-Three) ==
According to , the Gregorian calendar moves ahead $1$ day approximately every $3333$ years.
This is the result of using the slightly imprecise value of $365 \cdotp 2422$ days per tropical year.
Using the more precise value of $\approx 365 \cdotp 24219 \, 878$ days per tropical year, the number of years is approximately $3319$.
Note, however, that the Gregorian calendar moves backwards one day every $3319$ years, not ahead.
The origin of this statement is not given.
== Sources ==
* : $3333$
Category:Historical Notes
3334

== Number ==
$3334$ (three thousand,three hundred and thirty-four) is:
:$2 \times 1667$
:$3334^3 = 037 \, 059 \, 263 \, 704$; $0370 + 5926 + 3704 = 10 \, 000$
== Also see ==
* Squares of 3...34
* Cube of 3334
== Sources ==
* : $3334$
* : $3334$
Category:Specific Numbers
Category:3334
3337

== Number ==
$3337$ (three thousand, three hundred and thirty-seven) is:
:$47 \times 71$
:The $47$th second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $1962$, $2072$, $2185$, $2301$, $2420$, $2542$, $2667$, $2795$, $2926$, $3060$, $3197$:
::$3337 = \dfrac {46 \left({3 \times 47 + 1}\right)} 2$
== Also see ==
* 
Category:Specific Numbers
Category:3337
336

== Number ==
$336$ (three hundred and thirty-six) is:
:$2^4 \times 3 \times 7$
:The $14$th positive integer after $64$, $96$, $128$, $144$, $160$, $192$, $216$, $224$, $240$, $256$, $288$, $320$, $324$ with $6$ or more prime factors:
::$336 = 2 \times 2 \times 2 \times 2 \times 3 \times 7$
:The $28$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $262$, $268$, $276$, $288$, $290$, $292$, $304$, $306$, $322$, $324$, $326$
:The $32$nd highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $\ldots$, $120$, $144$, $168$, $180$, $210$, $216$, $240$, $288$, $300$:
::$\map {\sigma_1} {336} = 992$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:336
3364

== Number ==
$3364$ (three thousand, three hundred and sixty-four) is:
:$2^2 \times 29^2$
:The $47$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $2209$, $2304$, $2500$, $2809$, $2986$, $3136$, $3364$, $\ldots$
:The $58$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $1936$, $2025$, $2116$, $2209$, $2304$, $2401$, $2500$, $2601$, $2704$, $2809$, $2916$, $3025$, $3136$, $3249$:
::$3364 = 58 \times 58$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3364
3367

== Number ==
$3367$ (three thousand, three hundred and sixty-seven) is:
:$7 \times 13 \times 37$
:The $16$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$, $1105$, $1541$, $1729$, $1891$, $2465$, $2665$, $2701$, $2821$, $3281$:
::$3^{3367} \equiv 3 \pmod {3367}$
:The $34$th centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $2791$, $2977$, $3169$:
::$3367 = \ds 1 + \sum_{k \mathop = 1}^{34 - 1} 6 k = 34^3 - 33^3$
:The $37$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $2205$, $2356$, $2512$, $2673$, $2839$, $3010$, $3186$:
::$3367 = \ds \sum_{k \mathop = 1}^{37} \paren {5 k - 4} = \dfrac {37 \paren {5 \times 37 - 3} } 2$
:To multiply $3367$ by a $2$-digit integer $\sqbrk {xy}$, divide the $6$-digit integer $\sqbrk {xyxyxy}$ by $3$.
== Also see ==
* 3367 Multiplied by 2-Digit Number
* 
* 
* 
== Sources ==
* : $3367$
* : $3367$
Category:Specific Numbers
Category:3367
3367 Multiplied by 2-Digit Number

== Theorem ==
In order to multiply $3367$ by a $2$-digit integer $\sqbrk {xy}$:
:divide the $6$-digit integer $\sqbrk {xyxyxy}$ by $3$.
== Proof ==
We have that:
:$10101 = 3367 \times 3$
Then:
:$10101 \times \sqbrk {xy} = \sqbrk {xyxyxy}$
The result follows.
== Sources ==
* : Chapter $7$: Tricks of Lightning Calculators: Addendum
* : $3367$
* : $3367$
Category:Recreational Mathematics
Category:3367
337

== Number ==
$337$ (three hundred and thirty-seven) is:
:The $68$th prime number
:The smaller of the $5$th pair of primes whose prime gap is $10$:
::$347 - 337 = 10$
:The $6$th obstinate number after $1$, $127$, $149$, $251$, $331$
:The $8$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$, $337$
:The $17$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$, $157$, $167$, $179$, $199$, $311$
:The $17$th near-repdigit prime after $101$, $113$, $131$, $151$, $181$, $191$, $199$, $211$, $223$, $227$, $229$, $233$, $277$, $311$, $313$, $331$
:The $18$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $113$, $131$, $199$, $311$
:The $25$th long period prime after $7$, $17$, $19$, $23$, $29$, $\ldots$, $181$, $193$, $223$, $229$, $233$, $257$, $263$, $269$, $313$
:The $25$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $197$, $223$, $283$, $313$, $317$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:337
3375

== Number ==
$3375$ (three thousand, three hundred and seventy-five) is:
:$3^3 \times 5^3$
:The $3$rd power of $15$ after $(1)$, $15$, $225$:
::$3375 = 15^3$
:The $15$th cube number after $1$, $8$, $27$, $64$, $125$, $216$, $343$, $512$, $729$, $1000$, $1331$, $1728$, $2197$, $2744$:
::$3375 = 15 \times 15 \times 15$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3375
338

== Number ==
$338$ (three hundred and thirty-eight) is:
:$2 \times 13^2$
:The $54$th happy number after $1$, $7$, $10$, $13$, $19$, $23$, $\ldots$, $226$, $230$, $236$, $239$, $262$, $263$, $280$, $291$, $293$, $301$, $302$, $310$, $313$, $319$, $320$, $326$, $329$, $331$:
::$338 \to 3^2 + 3^2 + 8^2 = 9 + 9 + 64 = 82 \to 8^2 + 2^2 = 64 + 4 = 68 \to 6^2 + 8^2 = 36 + 64 = 100 \to 1^2 + 0^2 + 0^2 = 1$
== Also see ==
* 
::No further elements of this sequence will be tabulated on .
Category:Specific Numbers
Category:338
34

== Number ==
$34$ (thirty-four) is:
:$2 \times 17$
:The $2$nd non-square positive integer which cannot be expressed as the sum of a square and a prime:
::$10$, $34$, $\ldots$
:The $3$rd nontotient after $14$, $26$:
::$\nexists m \in \Z_{>0}: \map \phi m = 34$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $3$rd noncototient after $10$, $26$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 34$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The smallest positive integer which can be expressed as the sum of $2$ distinct lucky numbers in $4$ different ways:
::$34 = 1 + 33 = 3 + 31 = 9 + 25 = 13 + 21$
:The smallest positive integer which can be expressed as the sum of $2$ odd primes in $4$ ways:
::$34 = 31 + 3 = 29 + 7 = 23 + 11 = 17 + 17$
:The $4$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $\ldots$
:The $4$th heptagonal number after $1$, $7$, $18$:
::$34 = 1 + 7 + 11 + 16 = \dfrac {4 \paren {5 \times 4 - 3} } 2$
:The magic constant of a magic square of order $4$, after $1$, $(5)$, $15$:
::$\ds 34 = \frac 1 4 \sum_{k \mathop = 1}^{4^2} k = \frac {4 \paren {4^2 + 1} } 2$
:The $9$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$:
::$34 = 13 + 21$
:The $12$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$:
::$34 = 2 \times 17$
:The $24$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{34} = 17 \, 179 \, 869 \, 184$
== Also see ==
* Sum of 2 Lucky Numbers in 4 Ways
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
* 
== Sources ==
* : $34$
* : $34$
Category:Specific Numbers
Category:34
34,560

== Number ==
$34 \, 560$ (thirty-four thousand, five hundred and sixty) is:
:$2^8 \times 3^3 \times 5$
:The $5$th superfactorial after $1$, $2$, $12$, $288$:
::$34 \, 560 = 5\$ = 5! \times 4! \times 3! \times 2! \times 1!$
== Also see ==
* 
Category:Specific Numbers
Category:34,560
34,790

== Number ==
$34 \, 790$ (thirty-four thousand, seven hundred and ninety) is:
:$2 \times 5 \times 7^2 \times 71$
:The $23$rd integer $m$ such that $m! - 1$ (its factorial minus $1$) is prime:
::$3$, $4$, $6$, $7$, $12$, $14$, $30$, $32$, $33$, $38$, $94$, $166$, $324$, $379$, $469$, $546$, $974$, $1963$, $3507$, $3610$, $6917$, $21 \, 480$, $34 \, 790$
== Also see ==
* 
Category:Specific Numbers
Category:34,790
34,883,817,075

== Number ==
$34 \, 883 \, 817 \, 075$ is:
:$3^3 \times 5^2 \times 127 \times 503 \times 809$
:The larger of the $1$st of a set of $3$ amicable pairs whose sum is $64 \, 795 \, 852 \, 800$:
::$\map {\sigma_1} {29 \, 912 \, 035 \, 725} = \map {\sigma_1} {34 \, 883 \, 817 \, 075} = 64 \, 795 \, 852 \, 800 = 29 \, 912 \, 035 \, 725 + 34 \, 883 \, 817 \, 075$
== Also see ==
* 
Category:Specific Numbers
Category:34,883,817,075
340

== Number ==
$340$ (three hundred and forty) is:
:$2^2 \times 5 \times 17$
:The $14$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$, $320$, $340$
:The $33$rd noncototient after $10$, $26$, $34$, $50$, $\ldots$, $244$, $260$, $266$, $268$, $274$, $290$, $292$, $298$, $310$, $326$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 340$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:340
340,282,366,920,938,463,463,374,607,431,768,211,457

== Number ==
$340 \, 282 \, 366 \, 920 \, 938 \, 463 \, 463 \, 374 \, 607 \, 431 \, 768 \, 211 \, 457$ is:
:$59 \, 649 \, 589 \, 127 \, 497 \, 217 \times 5 \, 704 \, 689 \, 200 \, 685 \, 129 \, 054 \, 721$
:The $7$th Fermat number after ($3$), $5$, $17$, $257$, $65 \, 537$, $4 \, 294 \, 967 \, 297$, $18 \, 446 \, 744 \, 073 \, 709 \, 551 \, 617$:
::$340 \, 282 \, 366 \, 920 \, 938 \, 463 \, 463 \, 374 \, 607 \, 431 \, 768 \, 211 \, 457 = 2^{\paren {2^7} } + 1 = 2^{128} + 1$
:and the $3$rd not to be prime.
== Also see ==
* Prime Decomposition of 7th Fermat Number
* 
Category:Specific Numbers
Category:340,282,366,920,938,463,463,374,607,431,768,211,457
3400

== Number ==
$3400$ (three thousand, four hundred) is:
:$2^3 \times 5^2 \times 17$
:The $34$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $2296$, $2465$, $2640$, $2821$, $3008$, $3201$:
::$3400 = \ds \sum_{k \mathop = 1}^{34} \paren {6 k - 5} = 34 \paren {3 \times 34 - 2}$
== Also see ==
* 
Category:Specific Numbers
Category:3400
3403

== Number ==
$3403$ (three thousand, four hundred and three) is:
:$41 \times 83$
:The $82$nd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2701$, $2775$, $2850$, $2926$, $3003$, $3081$, $3160$, $3240$, $3321$:
::$3403 = \ds \sum_{k \mathop = 1}^{82} k = \dfrac {82 \times \paren {82 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3403
341

== Number ==
$341$ (three hundred and forty-one) is:
:$11 \times 31$
:The $1$st Poulet number:
::$2^{341} \equiv 2 \pmod {341}$: $341 = 11 \times 31$
:The $4$th Fermat pseudoprime to base $4$ after $15$, $85$, $91$:
::$4^{341} \equiv 4 \pmod {341}$
:The $11$th octagonal number, after $1$, $8$, $21$, $40$, $65$, $96$, $133$, $176$, $225$, $280$:
::$341 = 1 + 7 + 13 + 19 + 25 + 31 + 37 + 43 + 49 + 55 = \ds \sum_{k \mathop = 1}^n \paren {6 k - 5} = 11 \paren {3 \times 11 - 2}$
== Also see ==
* 
* 
* 
== Sources ==
* : $341$
* : $341$
Category:Specific Numbers
Category:341
342

== Number ==
$342$ (three hundred and forty-two) is:
:$2 \times 3^2 \times 19$
:The $12$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $148$, $189$, $235$, $286$:
::$342 = 1 + 7 + 11 + 16 + 21 + 26 + 31 + 36 + 41 + 46 + 51 + 56 = \dfrac {12 \paren {5 \times 12 - 3} } 2$
:The $29$th untouchable number after $2$, $5$, $52$, $88$, $96$, $120$, $124$, $\ldots$, $262$, $268$, $276$, $288$, $290$, $292$, $304$, $306$, $322$, $324$, $326$, $336$
== Also see ==
* 
* 
Category:Specific Numbers
Category:342
343

== Number ==
$343$ (three hundred and forty-three) is:
:$7^3$
:The $3$rd power of $7$ after $(1)$, $7$, $49$:
::$343 = 7^3$
:The $3$rd palindromic cube after $1$, $8$:
::$343 = 7^3$
:The $7$th cube number after $1$, $8$, $27$, $64$, $125$, $216$:
::$343 = 7 \times 7 \times 7$
:The $10$th Friedman number base $10$ after $25$, $121$, $125$, $126$, $127$, $128$, $153$, $216$, $289$:
::$343 = \paren {3 + 4}^3$
:The $18$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$, $119$, $170$, $210$, $214$, $217$, $265$, $282$, $310$, $322$, and the $2$nd of the only $2$ cubes, after $1$, whose divisor sum is square:
:
:The $31$st powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $\ldots$, $144$, $169$, $196$, $200$, $216$, $225$, $243$, $256$, $288$, $289$, $324$
== Also see ==
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:343
3432

== Number ==
$3432$ (three thousand, four hundred and thirty-two) is:
:$2^3 \times 3 \times 11 \times 13$
:The $7$th central binomial coefficient after $2$, $6$, $20$, $70$, $252$, $924$:
::$3432 = \dbinom {2 \times 7} 7 := \dfrac {14!} {\paren {7!}^2}$
== Also see ==
* 
Category:Specific Numbers
Category:3432
3435

== Number ==
$3435$ (three thousand, four hundred and thirty-five) is:
:$3 \times 5 \times 229$
:The $2$nd perfect digit-to-digit invariant after $1$:
::$3435 = 3^3 + 4^4 + 3^3 + 5^5$
== Also see ==
* 
== Sources ==
* : $3435$
* : $438,579,088$
* : $3435$
* : $438,579,088$
Category:Specific Numbers
Category:3435
3439

== Number ==
$3439$ (three thousand, four hundred and thirty-nine) is:
:$19 \times 181$
:The magic constant of a magic square of order $19$, after $1$, $(5)$, $15$, $34$, $\ldots$, $870$, $1105$, $1379$, $1695$, $2056$, $2465$, $2925$:
::$\ds 3439 = \dfrac 1 {19} \sum_{k \mathop = 1}^{19^2} k = \dfrac {19 \paren {19^2 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:3439
344

== Number ==
$344$ (three hundred and forty-four) is:
:$2^3 \times 43$
:The $34$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $244$, $260$, $266$, $268$, $274$, $290$, $292$, $298$, $310$, $326$, $340$:
::$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 344$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The numerator of an anomalously cancelling fraction:
== Also see ==
* Anomalous Cancellation: $\dfrac {3544} {7531}$
* 
Category:Specific Numbers
Category:344
345

== Number ==
$345$ (three hundred and forty-five) is:
:$3 \times 5 \times 23$
:The $15$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$, $100$, $126$, $155$, $187$, $222$, $260$, $301$:
::$345 = \dfrac {15 \paren {3 \times 15 + 1} } 2$
:The $19$th number after $1$, $3$, $22$, $66$, $70$, $81$, $94$, $115$, $119$, $170$, $210$, $214$, $217$, $265$, $282$, $310$, $322$, $343$ whose divisor sum is square:
:
:The $30$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $222$, $247$, $260$, $287$, $301$, $330$:
::$345 = \dfrac {15 \paren {3 \times 15 + 1} } 2$
:The $36$th sphenic number after $30$, $42$, $66$, $70$, $\ldots$, $246$, $255$, $258$, $266$, $273$, $282$, $285$, $286$, $290$, $310$, $318$, $322$:
::$345 = 3 \times 5 \times 23$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:345
346

== Number ==
$346$ (three hundred and forty-six) is:
:$2 \times 173$
:The $13$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$, $121$, $166$, $202$, $265$, $274$, $319$:
::$3 + 4 + 6 = 2 + 1 + 7 + 3 = 13$
:The $35$th noncototient after $10$, $26$, $34$, $50$, $\ldots$, $290$, $292$, $298$, $310$, $326$, $340$, $344$:
::$\nexists m \in \Z_{>0}: m - \map \phi m = 346$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:346
3465

== Number ==
$3465$ (three thousand, four hundred and sixty-five) is:
:$3^2 \times 5 \times 7 \times 11$
:The $5$th odd abundant number after $945$, $1575$, $2205$, $2835$:
::$\map {\sigma_1} {3465} - 3465 = 4023 > 3465$
== Also see ==
* 
Category:Specific Numbers
Category:3465
347

== Number ==
$347$ (three hundred and forty-seven) is:
:The $69$th prime number
:The larger of the $5$th pair of primes whose prime gap is $10$:
::$347 - 337 = 10$
:The upper end of the $6$th record-breaking gap between twin primes:
::$347 - 313 = 34$
:The $11$th Friedman number base $10$ after $25$, $121$, $125$, $126$, $127$, $128$, $153$, $216$, $289$, $343$:
::$347 = 7^3 + 4$
:The $13$th safe prime after $5$, $7$, $11$, $23$, $47$, $59$, $83$, $107$, $167$, $179$, $227$, $263$:
::$347 = 2 \times 173 + 1$
:The $18$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$, $157$, $167$, $179$, $199$, $311$, $337$
:The smaller of the $21$st pair of twin primes, with $349$
:The $26$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $197$, $223$, $283$, $313$, $317$, $337$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:347
3481

== Number ==
$3481$ (three thousand, four hundred and eighty-one) is:
:$59^2$
:The $48$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $2209$, $2304$, $2500$, $2809$, $2986$, $3136$, $3364$, $3481$, $\ldots$
:The $59$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $2304$, $2401$, $2500$, $2601$, $2704$, $2809$, $2916$, $3025$, $3136$, $3249$, $3364$:
::$3481 = 59 \times 59$
== Also see ==
* 
* 
Category:Specific Numbers
Category:3481
3486

== Number ==
$3486$ (three thousand, four hundred and eighty-six) is:
:$2 \times 3 \times 7 \times 83$
:The $42$nd hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $2145$, $2278$, $2415$, $2556$, $2701$, $2850$, $3003$, $3160$, $3321$:
::$3486 = \ds \sum_{k \mathop = 1}^{42} \paren {4 k - 3} = 42 \paren {2 \times 42 - 1}$
:The $83$rd triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $2701$, $2775$, $2850$, $2926$, $3003$, $3081$, $3160$, $3240$, $3321$, $3403$:
::$3486 = \ds \sum_{k \mathop = 1}^{83} k = \dfrac {83 \times \paren {83 + 1} } 2$
== Also see ==
* 
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Category:Specific Numbers
Category:3486
349

== Number ==
$349$ (three hundred and forty-nine) is:
:The $70$th prime number
:The lower end of the $7$th record-breaking gap between twin primes:
::$419 - 349 = 70$
:The larger of the $21$st pair of twin primes, with $347$.
== Also see ==
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Category:Specific Numbers
Category:349
35

== Number ==
$35$ (thirty-five) is:
:$5 \times 7$
:The $1$st positive integer $n$ such that $\map {\sigma_1} n = \dfrac {\map \phi n \times \map {\sigma_0} n} 2$:
::$\map {\sigma_1} {35} = 48 = \dfrac {\map \phi {35} \times \map {\sigma_0} {35} } 2$
:The $2$nd positive integer after $1$ whose square can be expressed as the sum of a sequence of odd cubes from $1$:
::$35^2 = 1225 = 1^3 + 3^3 + 5^3 + 7^3 + 9^3$
:The number of pairs of twin primes less than $10^3$
:The $4$th pentatope number after $1$, $5$, $15$:
::$35 = 1 + 4 + 10 + 20 = \dfrac {4 \paren {4 + 1} \paren {4 + 2} \paren {4 + 3} } {24}$
:The $4$th integer after $7$, $13$, $19$ the decimal representation of whose square can be split into two parts which are each themselves square:
::$35^2 = 1225$; $1 = 1^2$, $225 = 15^2$
:The $5$th pentagonal number after $1$, $5$, $12$, $22$:
::$35 = 1 + 4 + 7 + 10 + 13 = \dfrac {5 \paren {3 \times 5 - 1} } 2$
:The $5$th tetrahedral number, after $1$, $4$, $10$, $20$:
::$35 = 1 + 3 + 6 + 10 + 15 = \dfrac {5 \paren {5 + 1} \paren {5 + 2} } 6$
:The $5$th integer $n$ after $1$, $3$, $15$, $30$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:
::$\map {\sigma_0} {35} = 4$, $\map \phi {35} = 24$, $\map {\sigma_1} {35} = 48$
:The number of distinct free hexominoes
:The $9$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$:
::$35 = \dfrac {5 \paren {3 \times 5 - 1} } 2$
:The $13$th semiprime after $4$, $6$, $9$, $10$, $14$, $15$, $21$, $22$, $25$, $26$, $33$, $34$:
::$35 = 5 \times 7$
:The $18$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $33$, $35$, $\ldots$
:The $19$th after $1$, $2$, $4$, $5$, $6$, $8$, $9$, $12$, $13$, $15$, $16$, $17$, $20$, $24$, $25$, $27$, $28$, $32$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes
:The $25$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $20$, $21$, $24$, $25$, $26$, $27$, $30$, $31$, $32$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
:The $25$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
::$2^{35} = 34 \, 359 \, 738 \, 368$
:The maximum length of a non-crossing knight's tour on a standard chessboard.
:$35$ and $4374$ have the same prime factors between them as $36$ and $4375$:
::$35 = 5 \times 7$, $4374 = 2 \times 3^7$; $36 = 2^2 \times 3^2$, $4375 = 5^4 \times 7$
== Also see ==
* 35 Hexominoes
* Maximum Length of Non-Crossing Knight's Tour
* Prime Factors of 35, 36, 4734 and 4735
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== Sources ==
* : $35$
* : $35$
Category:Specific Numbers
Category:35
35,172

== Number ==
$35 \, 172$ (thirty-five thousand, one hundred and seventy-two) is:
:$2^2 \times 3^2 \times 977$
:The smaller of the $1$st pandigital pair, with $60 \, 984$, of integers each have squares which are themselves pandigital:
::$35 \, 172^2 = 1 \, 237 \, 069 \, 584$, $60 \, 984^2 = 3 \, 719 \, 048 \, 256$
== Also see ==
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== Sources ==
* : $57,321$
Category:Specific Numbers
Category:35,172
35,645

== Number ==
$35 \, 645$ (thirty-five thousand, six hundred and forty-five) is:
:$5 \times 7129$
:The $2$nd positive integer which is the sum of both $2$ and $3$ consecutive non-zero square numbers:
::$35 \, 645 = 1078^2 + 1079^2 + 1080^2 = 1321^2 + 1322^2$
== Also see ==
* 
== Sources ==
* : $365$
Category:Specific Numbers
Category:35,645
35,899

== Number ==
$35 \, 899$ (thirty-five thousand, eight hundred and ninety-nine) is:
:The $3814$th prime number
:The $6$th prime number after $5$, $19$, $101$, $619$, $4421$ of the form $\ds \sum_{k \mathop = 0}^{n - 1} \paren {-1}^k \paren {n - k}!$:
::$35 \, 899 = 8! - 7! + 6! - 5! + 4! - 3! + 2! - 1!$
== Also see ==
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Category:Specific Numbers
Category:35,899