Numbers in number theory

This page contains numbers appearing in number theory.
== List of numbers appearing in number theory ==
*2 is the only even .
*6 is the smallest .
*28 is the second .
*60 is the second .
*70 is the smallest .
*90 is the third .
*101 is the largest known in the form 10n+1.
*110 appears in the .
*132 is the 6th and a .
**It is also is the largest natural number n, such that πn is smaller than the first noncanonical -illion.
**132 is an even number, that has 12 divisors (1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132).
**132 is digit-reassembly number.https://en.wikipedia.org/wiki/Digit-reassembly_number It means that it is equal to the sum of all 2-digits numbers that can be made from the number itself (where the digits can't repeat in each 2-digits number): 12 + 13 + 21 + 23 + 31 + 32 = 132).http://oeis.org/search?q=132%2C264%2C396&language=english&go=Search It's also the smallest number with this property.
**Its prime factorization is 22 × 3 × 11.https://www.wolframalpha.com/input/?i=132
*145 appears in the 290 theorem.
*163 is the largest .
*165 is the larger of two known odd unitary superperfect numbers.
*The , which is used for finding the largest known primes, gives 194 after two iterations.
*203 appears in the 290 theorem.
*209 is the first composite .
*210 is the product of the single-digit s.
*231 is the 15th and the 21st .
**Since 21 is the 6th triangular number, 6 the 3rd triangular number, and 3 the 2nd triangular number, it also appears in the calculation of the triangrol, the triangoogol and the triangrolplex.
**It is also equal to 11!!!!.
**Furthermore, it is the number of es in a U.S. .
*272 is a and a pronic number.
**It is also the largest number n, such that 13n is smaller than a centillion.
*290 is the largest number in the 290 theorem, which is named for it.
*341 is the smallest to base 2.
*353 is the smallest number, whose fourth power is the sum of four smaller fourth powers.
**Some years in the have 353 days.
*385 is the 18th partition number and the 10th .
**Some years in the Hebrew calendar have 385 days.
*462 is the fifth largest known squarefree number of the form 2n−1Cn.
*496 (four hundred ninety-six) is the third . Its divisors are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496.
*561 is the first .
**561 is an interesting number primarily for the reason that 5612-5611-5610=314,159. The resulting number 314159 contains the first six decimal digits of .
**561 has 3, 11, and 17 as its prime factors, incidentally the sum of those prime factors is 31, which happens to be the first two decimal digits of pi.
*714 and 715 are a .Notable Properties of Specific Numbers (MROB) - 714
*777 is the 124th (in the mathematical sense).
*1,848 is the largest known .
*2,047 is the smallest composite Mersenne number with prime index, in this case, (211−1). The next Mersenne number however, which is 213−1 or 8,191, is prime.
**It is also the smallest to base 2.
**In the fast-growing hierarchy, it is equal to f2(8)−1 and f3(2)−1.
*The number 5,040 is the largest known factorial number which is the predecessor of a square number: 7! = 5,040 = 5,041−1 = 712−1.
*The number 5,041 is equal to 712. It is the largest known square number which is the successor of a factorial number.
*5,775 and 5,776 are the smallest pair of consecutive s.
*5,777 is the smallest odd composite .
*5,778 is the largest number that is both a triangle and a .
*5,993 is the largest known odd composite Stern number.
*8,128 (eight thousand one hundred twenty-eight) is the fourth perfect number.
*The 8,848th partition number is the largest one to be smaller than a googol.
**It is also the height of the in s.
*24,310 is the fourth largest known squarefree number of the form 2n−1Cn.
*65,537 is the largest known Fermat prime.OEIS, Sequence A019434. Accessed 2021-05-28.
*87,360 is the fourth unitary perfect number.
*92,378 is the third largest known squarefree number of the form 2n−1Cn.
*1,352,078 is the second largest known squarefree number of the form 2n−1Cn.
*The triple 2 + 6,436,341 = 6,436,343 is the with the highest known quality.
*33,550,336 (thirty-three millions five hundred fifty thousands three hundred thirty-six) is the fifth perfect number.
* has proven that there are infinitely many s not larger than 70,000,000.
**It is also the prize for correctly answering all sixteen questions in the Indian game show in s.
*1,766,319,049 was shown to be the least x for which 61x2+1=y2 for some y. This was shown by the Indian mathematician Chāskara in around 1200 AD.C. Selenius, [Rationale of the Chakravāla Process https://pdf.sciencedirectassets.com/272588/1-s2.0-S0315086000X00894/1-s2.0-0315086075901433/main.pdf] (p.179). Accessed 2021-05-28.
*5,425,069,447 and 5,425,069,448 are the smallest pair of consecutive s.
*8,589,869,056 is the sixth perfect number. Furthermore, it is the largest known perfect number .
*262,537,412,640,768,744 is an integer equal to 640,3203 + 744. It is almost equal to the Ramanujan constant.
**Its prime factorization is 23 × 3 × 10,939,058,860,032,031.
*221,256,270,138,418,389,602 is the largest known of the form 2n−1Cn.
**It is also equal to 72!/36!2/2.
**Its prime factorization is: 2 · 7 · 13 · 19 · 23 · 37 · 41 · 43 · 47 · 53 · 59 · 61 · 67 · 71, where · denotes multiplication.
*A number k≈4.63*10103 is given such that for n<k, there is only one such n where "\(\sigma_{od}(n)=\sigma_{od}(n+1)\) and neither n nor n+1 are perfect squares" holds, namely n=1KimIkikardesLiMa, On the problem σod(n)=σod(n+1) (2010) (p.557).
**The full value of k is 
== Figurate number collisions ==
This list contains numbers which are two types of s at the same time.
*36 is the 8th and the 6th .
*111 is a Hogben number, a and a .
*120 is the eighth and the 15th triangular number.
*133 is both a Hogben number and an .
**It is also an idoneal number.
**Furthermore, it is the first number n, such that πn is larger than the first noncanonical -illion.
**It’s a product of two different odd primes (7 and 19), so it’s an odd squarefree semiprime.https://www.wolframalpha.com/input/?i=133
**And the has 133.
*136 is a , a and the 16th triangular number.
**Therefore, it is the number of tiles in a double-15 set.
**It is also a constructible number.
**Since it is also approximately equal to the reciprocal of the fine structure constant, it was used in the definition of the Eddington number, which is equal to 136 × 2256.
**Furthermore, it is also the number of (namely, concourse, ground and 1 to 138; 41, 74, 110 and 137 have been skipped) served by the main service in the .
*175 is both a and a magic constant.
*176 is a , an octagonal number and a .
*210 is the largest number that is both a triangle and a .
**It is also a .
*232 is a cake number, a and a decagonal number.
**It is also the last number n, such that 20n is smaller than a centillion.
**And the isotope -232 is the longest-lived nuclide.
*252 is both a and the number of in a double-seven set.
*286 is both a and a tetrahedral number.
**It is also the smallest number n, for which 12n cannot be represented in the .
**There are exactly 286 primordial nuclides.
**And there is also with this number in the name.
*378 is a cake number, a and a triangular number.
*560 is both an octagonal number and a tetrahedral number.
*576 is both a cake number and a square number.
*697 is both a cake number and a heptagonal number.
**It is also the last number n, such that en is smaller than a centillion.
*870 is both a magic constant and a pronic number.
*946 is both a hexagonal pyramidal number and the number of tiles in a hypothetical double-42 domino set.
*969 is both a nonagonal number and a tetrahedral number.
*1,105 is a , a decagonal number and a magic constant.
*1,156 is both an and a square number.
*1,160 is both a cake number and an octagonal number.
*1,331 is both a and a .
*1,379 is both a magic constant and a central polygonal number.
*1,540 is the second largest number that is both a triangle and a tetrahedral number.
**It is also a decagonal number.
*1,716 is both a house number and a 6-simplex number.
*1,771 is both a central polygonal number and a tetrahedral number.
*1,794 is both a cake number and a nonagonal number.
*2,465 is both a magic constant and an octagonal number.
*2,626 is both a cake number and a decagonal number.
*2,925 is both a magic constant and a tetrahedral number.
*3,003 is the only known number larger than 1 which appears more than six times in Pascal's triangle.
*4,900 is the largest number that is both a square and a .
*4,960 is both a centered triangular number and a tetrahedral number.
*7,140 is the largest number that is both a triangle and a tetrahedral number.
**It is also the area of a in s.
*10,660 is both a decagonal number and a tetrahedral number.
*11,628 is the largest number that is both a triangle and a 5-simplex number.
*14,911 is both a and a magic constant.
*15,226 is both a cake number and a central polygonal number.
*19,600 is the largest number that is both a square and a tetrahedral number.
*19,669 is both a heptagonal number and a magic constant.
*24,310 is the largest number that is both a triangle and an 8-simplex number.
*47,972 is both a cake number and a pentagonal number.
*48,620 is both a 9-simplex and a pronic number.
*53,130 is both a 5-simplex and a pronic number.
*67,600 is both a cake number and a square number.
*208,335 is the largest number that is both a triangle and a pyramid number.
*226,981 is both a cube number and a .
*234,136 is both a nonagonal number and a tetrahedral number.
*428,536 is both a centered triangular number and a tetrahedral number.
*1,004,914 is both a cake number and a centered triangular number.
*1,175,056 is both a house number and a square number.
*1,414,910 is both a pronic number and a tetrahedral number.
*1,582,421 is both a and a house number.
*2,001,000 is both a hexagonal pyramidal number and the number of tiles in a hypothetical double-1999 domino set.
*2,287,231 is both a and a magic constant.
*2,635,752 is both a cake number and a pronic number.
*7,759,830 is a hexagonal number, a house number and a triangular number.
*9,653,449 is the largest number that is both a square and a .
*17,862,376 is both a cake number and a pentagonal number.
*90,525,801,730 is both a 31,265-agonal number and a 31,265-agonal pyramidal number. Numberphile, 90,525,801,730 Cannon Balls (accessed 2021-01-23)
The full version of Archimedes' cattle problem is partially related to figurate numbers as well.
*20,337,240 is both an octagonal number and a tetrahedral number.
*75,203,584 is both a cake number and a square number.
*319,118,031 is both a house number and a nonagonal number.
== Waring's problem-related numbers ==
This list contains numbers related to .
*138 is the number of known nonnegative integers which cannot be written as a sum of six nonnegative cubes; the largest of which is 8,042.
*223 is the only nonnegative integer which cannot be written as a sum of 36 nonnegative fifth powers.
**It is also the largest integer which cannot be written as a sum of 32, 33, 34 or 35 nonnegative fifth powers; there are only fifteen, ten, six and three nonnegative integers with this property, respectively.
**In some countries, such as China, the Band III ends at 223 MHz.
**It is also the number of non-control 8-bit characters.
*239 is the largest integer which cannot be written as a sum of eight nonnegative cubes; the only other nonnegative integer with this property is 23.
**It is also one of only seven nonnegative integers which cannot be written as a sum of eighteen fourth powers; the largest integer with this property is 559.
*241 is the number of known nonnegative integers which cannot be written as a ; the largest of which is 343,867.
*454 is the largest integer which cannot be written as a sum of seven nonnegative cubes; there are only 17 nonnegative integers with this property.
**It is also approximately the number of s in a .
**Furthermore, it is (not considering myriad, lakh and -illiard) the number of numbers with an accepted English name .
*466 is the largest integer which cannot be written as a sum of 28, 29, 30 or 31 nonnegative fifth powers; there are only 52, 41, 31 and 22 nonnegative integers with this property, respectively.
*559 is the largest integer which cannot be written as a sum of eighteen fourth powers; there are only seven nonnegative integers with this property.
*952 is the largest integer which cannot be written as a sum of 27 nonnegative fifth powers; there are only 66 nonnegative integers with this property.
*1,248 is the largest integer which cannot be written as a sum of seventeen fourth powers; there are only 31 nonnegative integers with this property.
*4,060 is the number of known nonnegative integers which cannot be written as a sum of five nonnegative cubes; the largest of which is 1,290,740.
*8,042 is the largest known integer which cannot be written as a sum of six nonnegative cubes; there are only 138 known nonnegative integers with this property.
*13,792 is the largest integer which cannot be written as a sum of sixteen fourth powers; there are only 96 nonnegative integers with this property.
*343,867 is the largest known integer which cannot be written as a sum of four tetrahedral numbers; there are only 241 known nonnegative integers with this property.
*1,290,740 is the largest known integer which cannot be written as a sum of five nonnegative cubes; there are only 4,060 known nonnegative integers with this property.
*113,936,676 is the number of known nonnegative integers which cannot be written as a sum of four nonnegative cubes; the largest of which is 7,373,170,279,850.
**Its factorization is 22 · 3 · 7 · 1,356,389.
*7,373,170,279,850 is the largest known integer which cannot be written as a sum of four nonnegative cubes; there are only 113,936,676 known nonnegative integers with this property.
**Its prime factorization is 2 × 52 × 18,521 × 7,961,957.
== Approximations of these numbers ==
For 8,042:
For 221,256,270,138,418,389,602:
== See also ==
*Figurate number collisions
*Prime numbers
*Mersenne-related numbers
*Mills' constant
*Wright's primes
== Sources ==
Category:Numbers
Category:Lists
Category:Class 1
Category:Class 2
Category:Unnamed numbers with 4 to 6 digits
Category:Unnamed numbers with 7 to 29 digits