## List of numbers appearing in number theory

• 2 is the only even prime number.
• 6 is the smallest perfect number.
• 28 is the second perfect number.
• 60 is the second unitary perfect number.
• 70 is the smallest weird number.
• 90 is the third unitary perfect number.
• 101 is the largest known prime in the form 10n+1.
• 110 appears in the 290 theorem.
• 132 is the 6th Catalan number and a pronic number.
• 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. 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). It's also the smallest number with this property.
• Its prime factorization is 22 × 3 × 11.
• 145 appears in the 290 theorem.
• 163 is the largest Heegner number.
• 165 is the larger of two known odd unitary superperfect numbers.
• The Lucas–Lehmer primality test, 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 Kummer number.
• 210 is the product of the single-digit prime numbers.
• 231 is the 15th partition number and the 21st triangular number.
• 272 is a constructible number 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 Fermat pseudoprime to base 2.
• 353 is the smallest number, whose fourth power is the sum of four smaller fourth powers.
• 385 is the 18th partition number and the 10th square pyramidal number.
• 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 perfect number. Its divisors are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496.
• 561 is the first Carmichael number.
• 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 pi.
• 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 Ruth–Aaron pair.
• 777 is the 124th lucky number (in the mathematical sense).
• 1,848 is the largest known idoneal number.
• 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.
• 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 abundant numbers.
• 5,777 is the smallest odd composite Stern number.
• 5,778 is the largest number that is both a triangle and a Lucas number.
• 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.
• 24,310 is the fourth largest known squarefree number of the form 2n−1Cn.
• 65,537 is the largest known Fermat prime.
• 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 abc triple with the highest known quality.
• 33,550,336 (thirty-three millions five hundred fifty thousands three hundred thirty-six) is the fifth perfect number.
• Yitang Zhang has proven that there are infinitely many prime gaps not larger than 70,000,000.
• 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 Ch&amacr;skara in around 1200 AD.
• 5,425,069,447 and 5,425,069,448 are the smallest pair of consecutive Achilles numbers.
• 8,589,869,056 is the sixth perfect number. Furthermore, it is the largest known perfect number not containing digit '4'.
• 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 squarefree number 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=1.
• The full value of k is
46,305,156,912,921,105,124,676,500,756,345,112,056,691,727,724,000,577,129,664,401,793,869,058,047,789,742,202,704,478,227,034,841,638,012

## Figurate number collisions

This list contains numbers which are two types of Figurate numbers at the same time.

• 36 is the 8th triangular number and the 6th square number.
• 111 is a Hogben number, a magic constant and a nonagonal number.
• 120 is the eighth tetrahedral number and the 15th triangular number.
• 133 is both a Hogben number and an octagonal number.
• 136 is a centered triangular number, a centered nonagonal number and the 16th triangular number.
• Therefore, it is the number of tiles in a double-15 domino 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 floors (namely, concourse, ground and 1 to 138; 41, 74, 110 and 137 have been skipped) served by the main service elevator in the Burj Khalifa.
• 175 is both a decagonal number and a magic constant.
• 176 is a cake number, an octagonal number and a pentagonal number.
• 210 is the largest number that is both a triangle and a pentatope number.
• 232 is a cake number, a central polygonal number and a decagonal number.
• It is also the last number n, such that 20n is smaller than a centillion.
• And the isotope thorium-232 is the longest-lived actinide nuclide.
• 252 is both a hexagonal pyramidal number and the number of pips in a double-seven domino set.
• 286 is both a heptagonal number and a tetrahedral number.
• 378 is a cake number, a hexagonal number 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 centered square number, a decagonal number and a magic constant.
• 1,156 is both an octahedral number and a square number.
• 1,160 is both a cake number and an octagonal number.
• 1,331 is both a cube number and a centered heptagonal number.
• 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 pyramid number.
• 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.
• 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 hex number 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 star number.
• 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 centered square number 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 centered pentagonal number 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 stella octangula number.
• 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. 

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 Waring's problem.

• 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 sum of four tetrahedral numbers; 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.
• 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:

Notation Approximation
Scientific notation $$8.042\times 10^3$$ (exact)
Arrow notation $$2\uparrow {13}$$
BEAF $$\{2,13\}$$
Fast-growing hierarchy $$f_2(9)<n<f_2(10)$$

For 221,256,270,138,418,389,602:

Notation Approximation
Scientific notation $$2.2126 \times 10^{20}$$
Arrow notation $$136↑10$$
BEAF $$\{136,10\}$$
Chained arrow notation $$136→10$$
Fast-growing hierarchy $$f_2(65)$$
Hardy hierarchy $$H_{\omega^2}(65)$$
Slow-growing hierarchy $$g_{\omega^{10}}(136)$$