nyuuzyou/wikis huggingface dataset:
15,760,111

== Number ==
$15 \, 760 \, 111$ is:
:The $1 \, 016 \, 632$nd prime number
:The $7$th of a sequence of $8$ prime numbers which have the same prime gaps as the $8$ primes starting at $11$
== Also see ==
* 
* 
Category:Specific Numbers
Category:15,760,111
15,760,117

== Number ==
$15 \, 760 \, 117$ is:
:The $1 \, 016 \, 633$rd prime number
:The $8$th of a sequence of $8$ prime numbers which have the same prime gaps as the $8$ primes starting at $11$
== Also see ==
* 
* 
Category:Specific Numbers
Category:15,760,117
15,841

== Number ==
$15 \, 841$ (fifteen thousand, eight hundred and forty-one) is:
:$7 \times 31 \times 73$
:The $9$th Carmichael number after $561$, $1105$, $1729$, $2465$, $2821$, $6601$, $8911$, $10 \, 585$:
::$\forall a \in \Z: a \perp 15 \, 841: a^{15 \, 840} \equiv 1 \pmod {15 \, 841}$
== Also see ==
* 
Category:Specific Numbers
Category:15,841
15,877

== Number ==
$15 \, 877$ (fifteen thousand, eight hundred and seventy-seven) is:
:The $1849$th prime number
:The $18$th prime $p$ such that $p \# - 1$, where $p \#$ denotes primorial (product of all primes up to $p$) of $p$, is prime:
::$3$, $5$, $11$, $13$, $41$, $89$, $317$, $337$, $991$, $1873$, $2053$, $2377$, $4093$, $4297$, $4583$, $6569$, $13 \, 033$, $15 \, 877$
== Also see ==
* 
* 
== Historical Note ==
== Sources ==
* 
* : $15,877 \# - 1$
Category:Specific Numbers
Category:15,877
15,877/Mistake

== Source Work ==
:
:The Dictionary
::$15,877 \# - 1$
== Mistake ==
:The largest prime of the form $p \# - 1$, which is also prime for $3$, $5$, $11$, $41$, $89$, $317$, $337$, $991$, $1873$, $2053$, $2377$, $4093$, $4297$, $4583$, $6569$, $13,033$. There are no more with $p \leqslant 35,000$.
$13$ is omitted from this list.
Interestingly, under the entry for $29$, does include $13$, but omits $337$.
== Sources ==
* : $15,877 \# - 1$
Category:Mistakes/Curious and Interesting Numbers
150,209

== Number ==
$150 \, 209$ (one hundred and fifty thousand, two hundred and nine) is:
:The $13 \, 867$th prime number
:The $22$nd integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$, $320$, $340$, $399$, $427$, $872$, $1477$, $6380$, $26 \, 951$, $110 \, 059$, $150 \, 209$
== Also see ==
* 
* 
Category:Specific Numbers
Category:150,209
1508

== Number ==
$1508$ (one thousand, five hundred and eight) is:
:$2^2 \times 13 \times 29$
:The $12$th heptagonal pyramidal number after $1$, $8$, $26$, $60$, $115$, $196$, $308$, $456$, $645$, $880$, $1166$:
::$1508 = 1 + 7 + 18 + 34 + 55 + 81 + 112 + 148 + 189 + 235 + 286 + 342 = \dfrac {12 \paren {12 + 1} \paren {5 \times 12 - 2} } 6$
== Also see ==
* 
Category:Specific Numbers
Category:1508
151

== Number ==
$151$ (one hundred and fifty-one) is:
:The $36$th prime number
:The $4$th near-repdigit prime after $101$, $113$, $131$
:The $7$th integer after $0$, $1$, $2$, $4$, $8$, $121$ which is palindromic in both decimal and ternary:
::$151_{10} = 12 \, 121_3$
:The $7$th prime number after $53$, $71$, $103$, $107$, $109$, $149$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$
:The $8$th palindromic prime after $2$, $3$, $5$, $7$, $11$, $101$, $131$
:The $9$th palindromic lucky number:
::$1$, $3$, $7$, $9$, $33$, $99$, $111$, $141$, $151$, $\ldots$
:The larger of the $12$th pair of twin primes, with $149$
:The $31$st lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $115$, $127$, $129$, $133$, $135$, $141$, $151$, $\ldots$
== Also see ==
* 
* 
* 
* 
* 
* 
* 
* 
Category:Specific Numbers
Category:151
1511

== Number ==
$1511$ (one thousand, five hundred and eleven) is:
:The $240$th prime number.
:The $21$st balanced prime after $5$, $53$, $157$, $173$, $211$, $257$, $263$, $373$, $563$, $593$, $607$, $653$, $733$, $947$, $977$, $1103$, $1123$, $1187$, $1223$, $1367$:
::$1511 = \dfrac {1499 + 1523} 2$
:The $51$st Sophie Germain prime after $2$, $3$, $5$, $11$, $23$, $29$, $41$, $\ldots$, $1031$, $1049$, $1103$, $1223$, $1229$, $1289$, $1409$, $1439$, $1451$, $1481$, $1499$:
::$2 \times 1511 + 1 = 3023$, which is prime.
:The $51$st near-repdigit prime after $101$, $113$, $131$, $151$, $\ldots$, $877$, $881$, $883$, $887$, $911$, $919$, $929$, $977$, $991$, $997$, $1117$, $1171$, $1181$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:1511
1512

== Number ==
$1512$ (one thousand, five hundred and twelve) is:
:$2^3 \times 3^3 \times 7$
:The $8$th positive integer after $504$, $756$, $806$, $1008$, $1148$, $1209$, $1472$ 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:
::$1512 = 24 \times 63 = 36 \times 42$
== Also see ==
* 
Category:Specific Numbers
Category:1512
1519

== Number ==
$1519$ (one thousand, five hundred and nineteen) is:
:$7^2 \times 31$
:The $23$rd centered hexagonal number after $1$, $7$, $19$, $37$, $61$, $91$, $127$, $\ldots$, $817$, $919$, $1027$, $1141$, $1261$, $1387$:
::$1519 = \ds 1 + \sum_{k \mathop = 1}^{23 - 1} 6 k = 23^3 - 22^3$
== Also see ==
* 
Category:Specific Numbers
Category:1519
152

== Number ==
$152$ (one hundred and fifty-two) is:
:$2^3 \times 19$
:The $20$th nontotient after $14$, $26$, $34$, $38$, $50$, $62$, $68$, $74$, $76$, $86$, $90$, $94$, $98$, $114$, $118$, $122$, $124$, $134$, $142$, $146$:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 152$
:where $\map \phi m$ denotes the Euler $\phi$ function
:The $9$th of the $17$ positive integers for which the value of the Euler $\phi$ function is $72$:
::$73$, $91$, $95$, $111$, $117$, $135$, $146$, $148$, $152$, $182$, $190$, $216$, $222$, $228$, $234$, $252$, $270$
== Also see ==
* 
* 
Category:Specific Numbers
Category:152
152,990

== Number ==
$152 \, 990$ (one hundred and fifty-two thousand, nine hundred and ninety) is:
:$2 \times 5 \times 15 \, 299$
:The $20$th 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:152,990
1520

== Number ==
$1520$ (one thousand, five hundred and twenty) is:
:$2^4 \times 5 \times 19$
:The $32$nd pentagonal number after $1$, $5$, $12$, $22$, $35$, $\ldots$, $925$, $1001$, $1080$, $1162$, $1247$, $1335$, $1426$:
::$1520 = \ds \sum_{k \mathop = 1}^{32} \paren {3 k - 2} = \dfrac {32 \paren {3 \times 32 - 1} } 2$
:The $63$rd generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $925$, $950$, $1001$, $1027$, $1080$, $1107$, $1162$, $1190$, $1247$, $1276$, $1335$, $1365$, $1426$, $1457$:
::$1520 = \ds \sum_{k \mathop = 1}^{32} \paren {3 k - 2} = \dfrac {32 \paren {3 \times 32 - 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1520
1521

== Number ==
$1521$ (one thousand five hundred and twenty-one) is:
:$3^2 \times 13^2$
:The $39$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $\ldots$, $625$, $676$, $729$, $784$, $841$, $900$, $961$, $1024$, $1089$, $1156$, $1225$, $1296$, $1369$, $1444$:
::$1521 = 39 \times 39$
:The $35$th positive integer which cannot be expressed as the sum of a square and a prime:
::$1$, $10$, $25$, $34$, $58$, $64$, $85$, $\ldots$, $706$, $730$, $771$, $784$, $841$, $1024$, $1089$, $1225$, $1255$, $1351$, $1414$, $1444$, $1521$, $\ldots$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1521
1523

== Number ==
$1523$ (one thousand, five hundred and twenty-three) is:
:The $241$st prime number.
:The $34$th safe prime after $5$, $7$, $11$, $23$, $47$, $\ldots$, $839$, $863$, $887$, $983$, $1019$, $1187$, $1283$, $1307$, $1319$, $1367$, $1439$, $1487$:
::$1523 = 2 \times 761 + 1$
:The $59$th left-truncatable prime after $2$, $3$, $5$, $7$, $\ldots$, $743$, $773$, $797$, $823$, $853$, $883$, $937$, $947$, $953$, $967$, $983$, $997$, $1223$, $1283$, $1367$, $1373$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1523
1525

== Number ==
$1525$ (one thousand, five hundred and twenty-five) is:
:$5^2 \times 61$
:The $25$th heptagonal number after $1$, $7$, $18$, $34$, $55$, $81$, $112$, $\ldots$, $874$, $970$, $1071$, $1177$, $1288$, $1404$:
::$1525 = \ds \sum_{k \mathop = 1}^{25} \paren {5 k - 4} = \dfrac {25 \paren {5 \times 25 - 3} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:1525
153

== Number ==
$153$ (one hundred and fifty-three) is:
:$3^2 \times 17$
:The sum of the first $5$ factorials:
::$153 = 1! + 2! + 3! + 4! + 5!$
:The $7$th Friedman number base $10$ after $25$, $121$, $125$, $126$, $127$, $128$:
::$153 = 51 \times 3$
:The $9$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $120$:
::$153 = 1 + 5 + 9 + 13 + 17 + 21 + 25 + 29 + 33 = 9 \paren {2 \times 9 - 1}$
:The $10$th pluperfect digital invariant after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and the $1$st non-trivial one:
::$1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153$
:The $17$th triangular number after $1$, $3$, $6$, $10$, $15$, $21$, $28$, $36$, $45$, $55$, $66$, $78$, $91$, $105$, $120$, $136$:
::$153 = \ds \sum_{k \mathop = 1}^{17} k = \dfrac {17 \times \paren {17 + 1} } 2$
:The $29$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
== Also see ==
* Repeated Sum of Cubes of Digits of Multiple of 3
* 
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== Historical Note ==
== Sources ==
* : $153$
* : $153$
Category:Specific Numbers
Category:153
153,846

== Number ==
$153 \, 846$ (one hundred and fifty-three thousand, eight hundred and forty-six) is:
:$2 \times 3^3 \times 7 \times 11 \times 37$
:Its cyclic permutations are multiples of $76 \, 923$:
::$153 \, 846 = 2 \times 76 \, 923$
::$615 \, 384 = 8 \times 76 \, 923$
::$461 \, 538 = 6 \times 76 \, 923$
::$846 \, 153 = 11 \times 76 \, 923$
::$384 \, 615 = 5 \times 76 \, 923$
::$538 \, 461 = 7 \times 76 \, 923$
== Also see ==
* Recurring Parts of Multiples of One Thirteenth
== Sources ==
* : $076,923$
* : $076,923$
Category:Specific Numbers
Category:153,846
153/Historical Note

== Historical Note on $153$ (One Hundred and Fifty-Three) ==
$153$ was the number of fish that the disciples of Jesus hauled out of the water when Jesus suggested they put their nets over the other side of the boat.
As a result, considerable numerological interpretation has been made of this.
For example, $153$ is the $17$th triangular number.
$17$ itself is the sum of $10$, for the Ten Commandments, and $7$, the number of the Gifts of the Holy Spirit.
And so on.
== Sources ==
* : $153$
* : $153$
Category:Historical Notes
1535

== Number ==
$1535$ (one thousand, five hundred and thirty-five) is:
:$5 \times 307$
:The $9$th Thabit number after $(2)$, $5$, $11$, $23$, $47$, $95$, $191$, $383$, $767$:
::$1535 = 3 \times 2^9 - 1$ 
== Also see ==
* 
Category:Specific Numbers
Category:1535
1537

== Number ==
$1537$ (one thousand, five hundred and thirty-seven) is:
:$29 \times 53$
:The $10$th Keith number after $14$, $19$, $28$, $47$, $61$, $75$, $197$, $742$, $1104$:
::$1$, $5$, $3$, $7$, $16$, $31$, $57$, $111$, $215$, $414$, $797$, $1537$, $\ldots$
== Also see ==
* 
Category:Specific Numbers
Category:1537
154

== Number ==
$154$ (one hundred and fifty-four) is:
:$2 \times 7 \times 11$
:The $12$th sphenic number after $30$, $42$, $66$, $70$, $78$, $102$, $105$, $110$, $114$, $130$, $138$:
::$154 = 2 \times 7 \times 11$
:The $21$st nontotient:
::$\nexists m \in \Z_{>0}: \phi \left({m}\right) = 154$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The $14$th noncototient after $10$, $26$, $34$, $50$, $52$, $58$, $86$, $100$, $116$, $122$, $130$, $134$, $146$:
::$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = 154$
:where $\phi \left({m}\right)$ denotes the Euler $\phi$ function
:The $6$th term of Göbel's sequence after $1$, $2$, $3$, $5$, $10$, $28$:
::$154 = \left({1 + 1^2 + 2^2 + 3^2 + 5^2 + 10^2 + 28^2}\right) / 6$
:The $30$th positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
:The $12$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
::$0$, $1$, $2$, $3$, $11$, $27$, $37$, $41$, $73$, $77$, $116$, $154$
== Also see ==
* 
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== Sources ==
* : $154$
Category:Specific Numbers
Category:154
154,345,556,085,770,649,600

== Number ==
$154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600$ is:
:$2^{15} \times 3^5 \times 5^2 \times 7^2 \times 11 \times 13 \times 17 \times 19 \times 31 \times 43 \times 257$
:The $1$st multiply perfect number of order $6$:
== Also see ==
* Smallest Multiply Perfect Number of Order 6
== Sources ==
* : $154,345,556,085,770,649,600$
Category:Specific Numbers
Category:154,345,556,085,770,649,600
1540

== Number ==
$1540$ (one thousand, five hundred and forty) is:
:$2^2 \times 5 \times 7 \times 11$
:The $4$th after $1$, $10$, $120$ of the $5$ tetrahedral numbers which are also triangular.
:The $20$th tetrahedral number, after $1$, $4$, $10$, $20$, $35$, $56$, $84$, $120$, $165$, $220$, $286$, $364$, $455$, $560$, $680$, $816$, $969$, $1140$, $1330$:
::$1540 = \ds \sum_{k \mathop = 1}^{20} \frac {k \paren {k + 1} } 2 = \dfrac {20 \paren {20 + 1} \paren {20 + 2} } 6$
:The $28$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $703$, $780$, $861$, $946$, $1035$, $1225$, $1326$, $1431$:
::$1540 = \ds \sum_{k \mathop = 1}^{28} \paren {4 k - 3} = 28 \paren {2 \times 28 - 1}$
:The $55$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $990$, $1035$, $1081$, $1128$, $1176$, $1225$, $1275$, $1326$, $1378$, $1431$, $1485$:
::$1540 = \ds \sum_{k \mathop = 1}^{55} k = \dfrac {55 \times \paren {55 + 1} } 2$
== Also see ==
* 
* 
* 
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== Sources ==
* : $1540$
* : $1540$
Category:Specific Numbers
Category:1540
1541

== Number ==
$1541$ (one thousand, five hundred and forty-one) is:
:$23 \times 67$
:The $6$th Fermat pseudoprime to base $5$ after $4$, $124$, $217$, $561$, $781$:
::$5^{1541} \equiv 5 \pmod {1541}$
:The $8$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$, $703$, $949$, $1105$:
::$3^{1541} \equiv 3 \pmod {1541}$
:The $23$rd octagonal number, after $1$, $8$, $21$, $40$, $65$, $\ldots$, $645$, $736$, $833$, $936$, $1045$, $1160$, $1281$, $1408$:
::$1541 = \ds \sum_{k \mathop = 1}^{23} \paren {6 k - 5} = 23 \paren {3 \times 23 - 2}$
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1541
1549

== Number ==
$1549$ (one thousand, five hundred and forty-nine) is:
:The $244$th prime number
:The $3$rd odd number after $1$, $5$ which cannot be expressed as the sum of an integer power and a prime number
== Also see ==
* 
* 
== Sources ==
* : $1549$
* : $1549$
Category:Specific Numbers
Category:1549
155

== Number ==
$155$ (one hundred and fifty-five) is:
:$5 \times 31$
:The $10$th second pentagonal number after $2$, $7$, $15$, $26$, $40$, $57$, $77$, $100$, $126$:
::$155 = \dfrac {10 \left({3 \times 10 + 1}\right)} 2$
:The $20$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $22$, $26$, $35$, $40$, $51$, $57$, $70$, $77$, $92$, $100$, $117$, $126$, $145$:
::$155 = \dfrac {10 \left({3 \times 10 + 1}\right)} 2$
:The $31$st positive integer $n$ such that no factorial of an integer can end with $n$ zeroes.
:The $59$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $95$, $96$, $102$, $108$, $114$, $119$, $120$, $125$, $143$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
== Also see ==
* 
* 
* 
* 
Category:Specific Numbers
Category:155
1552

== Number ==
$1552$ (one thousand, five hundred and fifty-two) is:
:$2^4 \times 97$
:The $32$nd second pentagonal number after $2$, $7$, $15$, $26$, $\ldots$, $610$, $672$, $737$, $805$, $876$, $950$, $1027$, $1107$, $1190$, $1276$, $1365$, $1457$:
::$1552 = \dfrac {32 \paren {3 \times 32 + 1} } 2$
:The $64$th generalized pentagonal number after $1$, $2$, $5$, $7$, $12$, $15$, $\ldots$, $925$, $950$, $1001$, $1027$, $1080$, $1107$, $1162$, $1190$, $1247$, $1276$, $1335$, $1365$, $1426$, $1457$, $1520$:
::$1552 = \dfrac {32 \paren {3 \times 32 + 1} } 2$
== Also see ==
* 
* 
Category:Specific Numbers
Category:1552
1556

== Number ==
$1556$ (one thousand, five hundred and fifty-six) is:
:$2^2 \times 389$
:The $9$th after $4$, $13$, $38$, $87$, $208$, $377$, $666$, $1027$ in the sequence formed by adding the squares of the first $n$ primes:
::$1556 = \ds \sum_{i \mathop = 1}^9 {p_i}^2 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 + 19^2 + 23^2$
== Also see ==
* 
Category:Specific Numbers
Category:1556
157

== Number ==
$157$ (one hundred and fifty-seven) is:
:The $37$th prime number
:The $1$st positive integer $n$ for which $\map \phi {2 n + 1} < \map \phi {2 n}$:
::$\map \phi {2 \times 157 + 1} = 144$, $\map \phi {2 \times 157} = 156$
:The $3$rd balanced prime after $5$, $53$:
::$157 = \dfrac {151 + 163} 2$
:The $8$th prime number after $53$, $71$, $103$, $107$, $109$, $149$, $151$ which cannot be expressed as either the sum of or the difference between a power of $2$ and a power of $3$
:The $9$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 8^2 + 29 = 157$ ()
:The $12$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$
:The $39$th and largest odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $\ldots$, $37$, $41$, $43$, $45$, $47$, $49$, $53$, $55$, $59$, $61$, $67$, $69$, $73$, $77$, $83$, $89$, $97$, $101$, $103$, $115$, $157$
== Also see ==
* 
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== Sources ==
* : $157$
Category:Specific Numbers
Category:157
1575

== Number ==
$1575$ (one thousand five hundred and seventy-five) is:
:$3^2 \times 5^2 \times 7$
:The $2$nd odd abundant number after $945$:
::$\map {\sigma_1} {1575} - 1575 = 1649 > 1575$
:With $1648$, an element of the $4$th quasiamicable pair:
::$\map {\sigma_1} {1575} = \map {\sigma_1} {1648} = 3224 = 1575 + 1648 + 1$
:The number of integer partitions for $24$:
::$\map p {24} = 1575$
:The $50$th Zuckerman number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $11$, $12$, $\ldots$, $1176$, $1184$, $1197$, $1212$, $1296$, $1311$, $1332$, $1344$, $1416$:
::$1575 = 9 \times 175 = 59 \times \paren {1 \times 5 \times 7 \times 5}$
== Also see ==
* 
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Category:Specific Numbers
Category:1575
158

== Number ==
$158$ (one hundred and fifty-eight) is:
:$2 \times 79$
:One of the factors of one of the terms of one of the $2$ representations of $635 \, 318 \, 657$ as the sum of $2$ fourth powers:
::$635 \, 318 \, 657 = 59^4 + 158^4$
:The $22$nd nontotient:
::$\nexists m \in \Z_{>0}: \map \phi m = 158$
:where $\map \phi m$ denotes the Euler $\phi$ function
== Also see ==
* 
* 
Category:Specific Numbers
Category:158
158,753,389,900

== Number ==
$158 \, 753 \, 389 \, 900$ is:
:$2^2 \times 5^2 \times 7^2 \times 17 \times 23 \times 41 \times 43 \times 47$
:The odds against being dealt a complete suit in a deal at bridge
== Also see ==
* Probability of Receiving Complete Suit as Hand at Bridge
== Sources ==
* : $158,753,389,900$
* : $158,753,389,900$
Category:Specific Numbers
Category:158,753,389,900
159

== Number ==
$159$ (one hundred and fifty-nine) is:
:$3 \times 53$
:The $2$nd positive integer after $79$ which cannot be expressed as the sum of fewer than $19$ fourth powers:
::$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$
:The $5$th Woodall number after $1$, $7$, $23$, $63$:
::$159 = 5 \times 2^5 - 1$
:The $6$th positive integer after $1$, $7$, $102$, $110$, $142$ the sum of whose divisors is a cube:
::$\map {\sigma_1} {159} = 216 = 6^3$
:The $32$nd lucky number:
::$1$, $3$, $7$, $9$, $13$, $15$, $21$, $\ldots$, $115$, $127$, $129$, $133$, $135$, $141$, $151$, $159$, $\ldots$
:The $61$st and last positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $124$, $136$, $137$, $141$, $142$, $147$ which cannot be expressed as the sum of distinct pentagonal numbers.
== Also see ==
* 
* 
* 
* 
* 
== Sources ==
* : $159$
* : $159$
Category:Specific Numbers
Category:159
1596

== Number ==
$1596$ (one thousand, five hundred and ninety-six) is:
:$2^2 \times 3 \times 7 \times 19$
:The $56$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $990$, $1035$, $1081$, $1128$, $1176$, $1225$, $1275$, $1326$, $1378$, $1431$, $1485$, $1540$:
::$1596 = \ds \sum_{k \mathop = 1}^{56} k = \dfrac {56 \times \paren {56 + 1} } 2$
== Also see ==
* 
Category:Specific Numbers
Category:1596
1597

== Number ==
$1597$ (one thousand, five hundred and ninety-seven) is:
:The $251$st prime number
:The $7$th Fibonacci prime after $2$, $3$, $5$, $13$, $89$, $233$.
:The $17$th Fibonacci number, after $1$, $1$, $2$, $3$, $5$, $8$, $13$, $21$, $34$, $55$, $89$, $144$, $233$, $377$, $610$, $987$:
::$1597 = 610 + 987$
:The $29$th of $29$ primes of the form $2 x^2 + 29$:
::$2 \times 28^2 + 29 = 1597$ ()
== Also see ==
* 
* 
* 
Category:Specific Numbers
Category:1597
159 is not Expressible as Sum of Fewer than 19 Fourth Powers

== Theorem ==
$159$ cannot be expressed as the sum of fewer than $19$ fourth powers:
:$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$
== Proof ==
We have:
:$4^4 = 256 > 159$
:$3^4 = 81$
:$2^4 = 16$
:$1^4 = 1$
Let us attempt to construct an expression of $159$ as the sum of fewer than $19$ fourth powers:
If no $3^4$ is used in our sum, the sum consists only of $2^4$ and $1^4$.
Using $2^4$ is more efficient than using $1^4$, since $2^4$ can replace $16 \times 1^4$.
So we have:
:$159 = 9 \times 2^4 + 15 \times 1^4$
which uses $24$ fourth powers.
If $3^4$ is used in our sum, the most efficient way is demonstrated above:
:$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$
which uses $19$ fourth powers.
We cannot use more than $2$ instances of $3^4$, as $2 \times 3^4 = 162 > 159$.
== Also see ==
* Hilbert-Waring Theorem
* Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers
* 319 is not Expressible as Sum of Fewer than 19 Fourth Powers
== Sources ==
* : $159$
* : $159$
Category:Fourth Powers
Category:Hilbert-Waring Theorem
Category:159